Samkhya: The Samkhya school believed that an unintelligent substance called Prakriti is the material cause of the Universe. Here is what Ishwara Krishna says in verse 16 of the Samkhya Karika:
kāraṇam asti avyaktaṃ pravartate triguṇataḥ samudayāt ca
pariṇāmataḥ salilavat pratiprattiguṇāśrayaviśeṣāt
The Unmanifest as the cause does exist. It functions through the three Attributes by combining and by modification, like water, due to the particular characteristics of the abode of each of the Attributes.
And here is what Gaudapada says in the Samkhya Karika Bhashya:
The well-known Unmanifest as the cause does exist from which proceed the mergent Mahat and the rest. Triguṇataḥ.—i.e., from (the Nature) possessed of the three Attributes. Triguṇa is that which has the three Attributes, viz., Sattva, Rajas and Tamas. Then, what is the sense?—(It is)—that the Nature is the equilibrium of Sattva, Rajas and Tamas. Again, by combining. As the three streams of the Ganges falling on the head of Rudra, (combine to) produce one stream, so the Unmanifest endowed with the three Attributes produces one Manifest. Or, as the combined threads produce cloth, so the Unmanifest creates Mahat and the rest, on account of the combination of the three Attributes. So, the manifested universe proceeds from the three Attributes and their combination. Well, as the entire Manifest proceeds from one Nature, so the former must be uniform.—This objection (does) not (arise). Because by modification, like water, due to the particular characteristics of abode of each of the Attributes. The three worlds born of one Nature are not of uniform nature. Gods are happy, men are unhappy and the animals are deluded. The Manifest, proceeding from one Unmanifest, becomes like water on account of modification based on the particular characteristics abiding in each of the three Attributes. The repetition of prati denotes “succession.” Guṇāśrayameans the abode of the Attributes;—its particular characteristic.—Taking- into consideration the particular characteristics of the abode of the three Attributes, the Manifest is created from the modification based upon the particular characteristics of the abode of the three Attributes. For example, water falling from the sky is of uniform nature; but coming into contact with different forms, that water becomes of different forms, due to different tastes. Similarly, the three worlds proceeding from one Nature, are not of uniform nature. Among the gods, the Sattva is predominant, and Rajas and Tamas are indifferent; so they are very happy. Among men, the Rajas is predominant, and Sattva and Tamas are indifferent; therefore, they are very unhappy, Among animals the Tamas is predominant, and Sattva and Rajas are indifferent; so they are very insensible.
Vaisheshika: The Vaisheshika school also believed that atoms are the material cause of the Univese, and this was the central doctrine of the Vaisheshika school, in contrast to the Nyaya school which didn't really emphasize this concept. Here is what Adhyaya 4 Pada 1 Sutra 2 of the Vaisheshika Sutras:
tasya kārya liṅgam
The effect is the mark (of the existence) of the ultimate atom.
Here is what Shankara Mishra says in the Upaskara:
Now the interrelation of parts and wholes is perceived. If it were unlimited, there would be no difference in size of measurement between mount Meru and a grain of mustard seed; for, they wauld be without distinction, both being orginated by infinite parts. Nor can it be said that difference will be caused by the differences of the size of each part, and of the aggregation of parts; for, without a difference of number, these also would be impossible. If it be said that pralaya or destruction of the creation may be the limit (of the series of parts and wholes) ; (we reply that) the final something ex-hypothesi) having no parts, pralaya itself would be impossible, for it is only disjunction and destruction of parts which can destroy substances. Nor is disjunction the limit, for it is impossible for it to have only one substratum. Therefore, a substance without parts, must be the limit, and this the ultimate atom. A mote is not the limit ; for, being a visible substance, it possesses magnitude, and is composed of more substance than one; magnitude, as the cause of visual percep tion, presupposes, or is dependent upon, multiplicity of substance; else there would be no magnitude even, what then would be the cause? Nor are the constituent parts of the mote atoms, for we must infer that they also, as originative of a substance possessing magnitude,. are compossed of parts, like thread, and like potsherd. Therefore. whatever substance is an effect, is composed of parts, and whatever substance is composed of parts, is an effect. So that from whichever part the nature of being an effect goes away, from it goes away also the nature of being made up of parts. This is the proof of the existence of indivisible ultimate atoms.
I'm reminded of the Banach-Tarski paradox.
Purva Mimamsa: The Purva Mimamsa school also believed that atoms are the material cause of the Universe, but the two sub-schools of Purva Mimamsa, the Bhatta school and the Prabhakara school had slightly different views on the nature of atoms. The Bhatta school, which followed the Purva Mimamsa philosopher Kumarila Bhatta, believed that atoms are perceptible. You know those dust particles you see when a ray of sunlight shines through a window? The Bhatta school thought those were atoms. Here is what Narayana Pandita says in the Manameyodaya:
These five are substances with parts. The parts are the primal atoms.... Those particular minute entities shining in the sun-beam proceeding through the opening of a window, these then the disciples of Kumarila, indeed, assume as the primal atoms in this connection, because there is no means for the valid knowledge of anything beyond these, and because there is no need for the assumption of anything beyond these.
This is in contrast to the Nyaya and Vaisheshika schools, which believed that atoms are imperceptible, and that those dust particles seen in a ray of sunlight shining through a window are not atoms, but triads, which consist of three dyads, each of which consist of two atoms. The Prabhakara school agreed with the Nyaya and Vaisheshika schools and disagreed with the Bhatta school on this issue. Here is what A. Rammulu says in his book "A Study of Differences between Bhatta and Prabhakara Schools (Mimamsa)":
The Prabhakara, in agreement with the Vaisheshika, accepts atoms as the ultimate constituents of the physical universe. These are indivisible and eternal. He accepts the existence of atoms on the basis of inference as follows: "Triad is a composite of parts (three dyads), because it is a perceptible substance, like a cloth." and "Dyad also is a composite of parts, because it produces large magnitude, i.e. triad, like a thread." Further the atom should not be divided into parts, because that would lead to the defect of ad infinitum.
Vedanta: Unlike the other Astika schools, which believed that the Universe was made of unintelligent substances like Prakriti or atoms, most members of the Vedanta school believe that the material cause of the Universe is an intelligent being, namely Brahman. Here is what Vyasa says in Adhyaya 1 Pada 4 of the Brahma Sutras:
Topic-7: Brahman as Material Cause
23. Brahman must be the material cause as well, so as not to contradict the proposition and the illustration.
24. This is also understood from the teaching about the wilt to create.
25. And because both (origin and dissolution) are taught directly (from Brahman).
26. (Brahman is the material cause) because of action related to Itself by way of change of form.
27. And because Brahman is declared to be the source (yoni).
Here is what Adi Shankaracharya says in his Brahma Sutra Bhashya:
Brahman is the material cause for that reason also that Scripture--in the passage, 'That made itself its Self' (Taitt. Up. II, 7)--represents the Self as the object of action as well as the agent.--But how can the Self which as agent was in full existence previously to the action be made out to be at the same time that which is effected by the action?--Owing to modification, we reply. The Self, although in full existence previously to the action, modifies itself into something special, viz. the Self of the effect. Thus we see that causal substances, such as clay and the like, are, by undergoing the process of modification, changed into their products.--The word 'itself' in the passage quoted intimates the absence of any other operative cause but the Self. The word 'parinâmât' (in the Sûtra) may also be taken as constituting a separate Sûtra by itself, the sense of which would be: Brahman is the material cause of the world for that reason also, that the sacred text speaks of Brahman and its modification into the Self of its effect as co-ordinated, viz. in the passage, 'It became sat and tyat, defined and undefined' (Taitt. Up. II, 6).
And here is what Ramanujacharya says in the Sri Bhashya:
This means--owing to the essential nature of modification (parinâma). The modification taught in our system is not such as to introduce imperfections into the highest Brahman, on the contrary it confers on it limitless glory. For our teaching as to Brahman's modification is as follows. Brahman--essentially antagonistic to all evil, of uniform goodness, differing in nature from all beings other than itself, all-knowing, endowed with the power of immediately realising all its purposes, in eternal possession of all it wishes for, supremely blessed--has for its body the entire universe, with all its sentient and non-sentient beings--the universe being for it a plaything as it were--and constitutes the Self of the Universe. Now, when this world which forms Brahman's body has been gradually reabsorbed into Brahman, each constituent element being refunded into its immediate cause, so that in the end there remains only the highly subtle, elementary matter which Scripture calls Darkness; and when this so-called Darkness itself, by assuming a form so extremely subtle that it hardly deserves to be called something separate from Brahman, of which it constitutes the body, has become one with Brahman; then Brahman invested with this ultra-subtle body forms the resolve 'May I again possess a world-body constituted by all sentient and non-sentient beings, distinguished by names and forms just as in the previous aeon,' and modifies (parinâmayati) itself by gradually evolving the world-body in the inverse order in which reabsorption had taken place.
All Vedânta-texts teach such modification or change on Brahman's part. There is, e.g., the text in the Brihad-Âranyaka which declares that the whole world constitutes the body of Brahman and that Brahman is its Self. That text teaches that earth, water, fire, sky, air, heaven, sun, the regions, moon and stars, ether, darkness, light, all beings, breath, speech, eye, ear, mind, skin, knowledge form the body of Brahman which abides within them as their Self and Ruler. Thus in the Kânva-text; the Mâdhyandina-text reads 'the Self' instead of 'knowledge'; and adds the worlds, sacrifices and vedas. The parallel passage in the Subâla-Upanishad adds to the beings enumerated as constituting Brahman's body in the Brihad-Âranyaka, buddhi, ahamkâra, the mind (kitta), the Un-evolved (avyakta), the Imperishable (akshara), and concludes 'He who moves within death, of whom death is the body, whom death does not know, he is the inner Self of all, free from all evil, divine, the one god Nârâyana."
It should be noted that there's one Vedantic philosophy, the Dvaita philosophy of Madhvacharya, which disagrees with the rest of the Vedanta school, instead agreeing with the view of the Samkhya and Yoga schools that Prakriti is the material cause of the Universe. See my answer here for how Madhvacharya interprets various Sutras in the Brahma Sutras which other commentators interpret as saying that Brahman is the material cause of the Universe.
Also, see my answer here for how the Vedanta school refutes the views that Prakriti or atoms are the material cause of the Universe.