First, in his translation, Prabhupāda is trying to reach a broad foreign audience not familiar with Vedantic terms. Rather than translate ‘tamah’ as ‘mode of ignorance’ it is better to just use the word tamas or tamah – one of the three gunas - qualities. Oversimplification has its merits – and demerits. His translation is a little confusing as he translates ‘tamah’ as ‘ignorance’ and ‘ajñāna-jam’ as ‘produced of ignorance’. Two different Sanskrit words with different purports but ONLY apparently almost the same meaning in English. Tamah is the guna known as tamah or tamas. ‘ajñāna-jam’, however, means ‘produced/born of ignorance’ – but all the gunas (sattwa, rajas, and tamas) are ‘ajñāna-jam’ – not only tamas. Several translators have used similar translations of ‘ajñāna-jam’ - I do not mean to single out Prabhupāda - BUT it should not be equated to mean ONLY that out of which tamas is produced. Remember that a more literal translation of ‘jnana’ is ‘knowledge’ and ‘ajnana’ - ‘not’ knowledge. The juxtaposition of the English word ignorance in the same sentence inclines the mind to see the same meaning to the word. A more literal translation would be – ‘the guna known as tamas produced out of that which is not knowledge’. Which leads to what is meant by knowledge and not-knowledge?
The qualities/definitions of the three gunas are clearly stated in Chapter 14. Where the gunas are derived from is stated in Gita 13.19 (“…all forms and gunas are born of Prakriti”).
A clearer understanding of all of this can be had from Sadananda’s Vedantasara verse 34 (original Sanskrit/English text with commentary downloadable here - https://www.scribd.com/doc/79765830/Vedantasara-of-Sadananda-translated-with-commentary-of-Swami-Nikhilananda-of-Ramakrishna-Order-1931) It says:
But ignorance [meaning not Knowledge] is described as something positive though intangible, which cannot be described either as being or non-being, which is made of three qualities [Gunas] and is antagonistic to Knowledge. Its existence is established from such experiences as “I am ignorant”, and from such Sruti passages as, “The power belonging to God Himself, in hidden its own qualities” (Svet. Up. I. 3).
In his footnotes to the above, Swami Nikhilananda says:
But – The text offers a special theory on the subject.
Something – This word has a special significance. It is not used to denote its indescribable nature, nor its antagonism to Knowledge and Truth as these ideas have been well expressed by separate phrases. Its special significance is to posit ignorance as the source or cause of illusion.
Positive – This is a difficult word and requires some explanation. This part of the definition is given in order to refute the contention that ignorance is mere negation, as it is antagonistic to Knowledge. The Nyaya school says that the absence of Knowledge is ignorance and so it is a negation. But the Vedantist says it is not a negation. He asks, what is Knowledge whose negation is contended to be ignorance? We can understand Knowledge from three aspects. Firstly, Knowledge is used as synonymous with ‘the Witness, the Perceiver,” (Svet. Up. 6. 11). Its absence cannot be called ignorance as it is eternal and therefore can never be associated with a state of negation. Secondly, a particular function of mind is termed knowledge, as in the passage, “Through understanding one understands the Rig-Veda” (Chh. Up. 7. 7. 1). But here ‘knowledge’ is used in only in an indirect sense. No mental function can illumine an object unless it has the Self at its back. The eyes, ears, etc., seem to perform their functions consciously because they draw their consciousness from the Self. Comp. – “All this is guided by Consciousness and is based on Consciousness; this universe has Consciousness for its guide, Consciousness for its base; Consciousness is Brahman.” (Ait. Up. 5. 3). Hence, under no circumstances can this Knowledge exist in a negative state. Thirdly, ignorance cannot be said to consist of the negation of knowledge, particular or general. Because when a man makes a statement as, “I am ignorant, I do not know anything,” even then he does not lose all sense of perception. Though he may not perceive a particular object, he perceives another. Again, there cannot be any negation of general knowledge as without it knowledge of a particular object becomes impossible. Therefore it stands to reason that Knowledge which is eternal, ever-existent and positive can never be connected or associated with negation. But when ignorance is said to be (positive), it does not denote an absolute substance which only Brahman is. Were it so, there would not be any liberation. Therefore this term is used to differentiate it from negation. Ignorance is different from reality and unreality, as neuter is different from male and female. Really this ignorance can never be properly explained. It has found a place in Vedanta philosophy in order to explain the otherwise inexplicable production of the phenomenal world. It is absurd to seek for its proof. It cannot be roved by our reasoning because human reasoning can never be free from (ignorance). To prove it by reasoning is like seeing darkness with the help of darkness. Nor can it be proved by Knowledge, as at the awakening of Knowledge there cannot remain any trace of ignorance. To prove ignorance by Knowledge is like seeing darkness by a blazing light. Comp. – “The characteristic of ignorance is its very unintelligibleness. It cannot bear any proof, or it will be a real thing” (Brihadaranyakavartika verse 181). – “This illusion is without support and contradictory to all reasoning. It cannot bear any reasoning just as darkness cannot stand the sun.” Therefore like the fanciful imagination of the sun by one who is blind by day, the sages called ignorance indescribable, as it is neither real nor unreal, nor real-unreal, neither with parts nor without parts, and neither separable from Knowledge nor inseparable from It.
Which etc. – It is not existent because it disappears at the awakening of Knowledge. It is not non-existent like a child of a barren woman otherwise there would not be any dealing with the world.
Three qualities – These are Sattva (serenity), Rajas (activity) and Tamas (inertia). These qualities have been mentioned in the following scriptural passages, though the Sankhya philosophy may justly claim to have fully developed the theory of the three Gunas.
“There is one unborn (feminine) being, red, white, and black, producing manifold offspring of the same nature as itself” (Svet. Up. 4. 5). This refers to (Nature) which is composed of the three qualities mentioned above. These three qualities are found to exist in all the products of Prakriti. Comp. – “The red colour that we notice in fire is its own colour, the white colour the colour of water, and the black colour is the colour of earth” (Chh. Up. 6. 4. 1). Like its effects, the cause, which is ignorance, is also made of the three qualities, though in the latter case they remain in an unmanifested state. Though the three qualities are attributes of (ignorance), yet they are its essential parts, as substance is inseparable from its attributes.
Antagonistic etc. – This definition is given in order to refute the contention that there would never be any cessation of phenomena, as the eternal ignorance manifested in the form of sky etc., appears to be real. Ignorance with all its modifications vanishes away at the dawn of Knowledge.
I am etc. – This experience illustrates the positiveness of ignorance.
Such Sruti etc. – Comp. – “Knowledge is enveloped by ignorance, hence the creatures are deluded” (Gita 5. 15). – “Veiled by this illusion owing to the association of Gunas, I am not manifest to all” (Gita 7. 25).
You can also see by the above quote, the answer to your question – What vedic scripture is Prabhupāda referring to? As stated above, although the gunas were fully developed in the Samkhya philosophy, there are references to the three gunas in the Chandogya and Svetasvatara Upanishads. But, in the Upanishads they are not referred to as the ‘gunas’, but as colors. The term gunas came after the Upanishads. You can also see from the above quote a discussion of the term ignorance meaning ‘not-knowledge’ which lends itself to a better understanding of ajñāna-jam in the Gita text 14.8.