Among the six darshanas how many of them subscribe to vedas and directly mention about vedas or hold vedas as authority?
All of them accept the existence of Atman (Jiva) and Brahman (God) and all of them accept the validity/authenticity of Vedas.
Refer following the closely related QA by Keshavan for detailed information:
- How do different Astika schools prove the existence of the soul?
- How do different Astika schools prove the existence of God?
Touching the point of question, I am here citing Darshana Sutras to show how each of them accept the authenticity of Vedas. Let's take different schools one by one:
tadvacanādāmnāyasya prāmāṇyam ||3||
Because it is an exposition of that, it (this teaching) has the authority of Veda
vaidikaṃ ca ||10||
5.2.10 and it conforms with the Veda.
Sometimes Vedas are also mentioned by alternative terms "Shruti" and "Aagama" in Darshana Sutras. Vaisheshiki Sutras 2.1.17 and 3.2.8 are also a proof that Vaisheshika Darshana accept the authenticity of Vedas.
2.1.69 The Veda is reliable like the spell and medical science, because of the reliability of their authors
Note that Yoga darshana is based on Samkhya of which views on Vedas are already presented.
1.7. Right knowledge is inference, tradition and genuine cognition.
1.49. The wisdom obtained in the higher states of consciousness is different from that obtained by inference and testimony as it refers to particulars.
Here the words "genuine cognition" and "testimony" refers to nothing other than Vedas since Vedas are raveled by Rishis in deep meditation. Read Swami Vivekananda's explanation in RajaYoga Chapter - I for more justification.
Vedas in in the name of this Darshana which is primarily based on Upanishads.
1.1.3 (The omniscience of Brahman follows) from its being the source of Scripture.
1.1.11 and because it is directly stated in Scripture (therefore the all-knowing Brahman is the cause of the world).
Here scriptures means Vedas and Upanishads. Refer commentary by Adi Shankaracharya for more information.
No other school has defended for the authority of Vedas as logically as Mimansa school. Instead of reproducing the citations, let me refer to a very useful explanation with citation by Tezz on What is the validity of Authoritativeness of vedas?.
Hence All the Aasktika philosophy accept Vedas as authentic.
Swami Vireswarananda in his Introduction to his translation of the Brahma Sutras writes on page ii-iii (here - https://www.wisdomlib.org/hinduism/book/brahma-sutras/d/doc62756.html):
The destructive criticism of everything in the old system by the Chârvâkas and others set the orthodox section to organize their belief on a more rationalistic basis and render it immune against all such criticism. This led to the foundation of the six systems of orthodox Hindu philosophy—orthodox in the sense that they accepted the authority of the Vedas in things transcendental—while there were others who did not accept this authority and therefore were dubbed heterodox, though otherwise they too were the outcome of Upanishadic thought. The acceptance of the authority of the Vedas by these orthodox schools, however, does not mean that they accepted them in toto . Their allegiance to the Vedas varied widely and often it was too loose. Of the six orthodox schools, viz. Nyâya, Vaiseshika, Sânkhya, Yoga, Purva Mimâmsâ and Uttara Mimâmsâ or Vedanta, the last two are intimately connected with the Vedas, which is one of the reasons why they are not mentioned in the Jaina and Buddhistic literature, while the others are mentioned.
These six orthodox systems of thought developed side by side at different intellectual centres, of which there were a good number all over the country even during the Upanishadic period. Again in each system there were shades of difference. Thus for centuries philosophic thought developed in India till at last it became so unwieldy that a regular systematization of each school of thought was found a great necessity. This led to the Sutra literature.
author's footnote  - Âstika (orthodox) and Nâstika (heterodox) had nothing to do with belief or non-belief in the existence of a God. Sânkhya and Mimâmsâ which did not accept an Iswara were yet regarded Âstika (orthodox).
All of the six systems accept the vedas as authority and are astika. See Sharma's A Critical Survey of Indian Philosophy (here - https://archive.org/details/IndianPhilosophyACriticalSurvey). Here are a few references in his book.
Nyâya - Accept the authority of the vedas. Chapter titled Nyaya, section VIII Soul.
Vaiseshika - Believes in the authority of the Vedas. Chapter titled Vaiseshika, section XI God.
Sânkhya - Believes in the authority of the vedas. See Chapter titled Sankhya, section VIII God.
Yoga - called theistic Sankhya, accepts the vedas.
Purva Mimâmsâ - accept the vedas.
Uttara Mimâmsâ - accept the vedas.