I am answering for the time being only from Rig Veda. The answer may be updated in future with references from the other Vedas.
First of all, a Rishi is a person to whom is revealed at least one Rik mantra of the Rig Veda. The Rishi literally hears the mantra by super- physical methods in a state like trance, which he has achieved by means of tapas.
Now, the source of all the Rik mantras has already been declared in a Rig Veda Mandala 1 mantra as follows:
Rcho akshare parame vyoman (1) Yasmin devA adhi visve nisheduh (2)
yastan na veda kim rchA karishyati (3)
Riks exist in a supreme ether, imperishable and immutable (1), in
which all the Gods are seated (2) ; One who knows not that, what shall
he do with the RIk ? (3)
This clearly indicates that Riks are not products of a human mind. Their source is the higher ether in which all the Gods are seated. This proves the apaureshaya nature of Veda mantras.
However, the full-fledged apaurasheya doctrine would also require that the Rishi seeing the whole or full form of the mantra .
Now, this idea is not validated in Rig Veda. The revelations that are received by the Rishis come from a higher plane alright but the Rishis also transcribe these revelations in their heart.
Eshah vah stomo maruto namsvan (1), hridA tashto manasAdhyAyi devAh
O Maruts, the hymn of your affirmation (stoma), is fraught with my
obeisance (1), It was framed by the heart, it was established by the
mind, O ye Gods (2).
RV 1.67.4 also states that the mantra is crafted or formed (tashtam) in the heart.
RV 1.171.2 ----> Rig Veda Samhita, Mandala 1, Sukta 171, Mantra 2.