Kindly note that Mimamsa school does not admit of a single supreme God (Ishvara) and, therefore, the question of the Vedas being authored by Ishvara does not arise. Thus, the question is now limited to whether humans authored the Vedas.
Sabara's bhashya on Jaimini's Mimansa Sutras give long explanation on why the Vedas are eternal and authorless.
Shabdas of two varieties: apaurusheya (Impersonal) and paurusheya (personal). The Vedas are the former. Mimamsakas argue that that the Vedas are svata-pramanya (intrinsically valid) and because the sabda is eternal, the vedas are eternal.
Their argument relies on an epistemological theory by which all utterances are valid. All utterances are assumed to be true unless falsified by evidence. The validity of a human utterance; however, is questionable because it could be falsified later.
From Sabara Bhashya 1.1.4
The cognition by a person is only by sense organs and it is not a means of dharma, as these organs are incapable of knowing dharma [in the matters of beyond time and perception].
The previous set of text in the bhashya 1.1.2 explains it in more detail.
It is the Vedic injunction that is capable of knowing what is past,
present, future and also what is subtle, hidden or remote. This cannot
be from sensory perception….If the words emanate from a trustworthy
person, it must be true; if it emanates from an untrustworthy person,
it is unreliable. However, in matters that cannot be perceived
directly by the senses, all words from men are unreliable as they
emanate from the mind of a human being. Dharma is not perceptible by
senses and it is only by means of words. For these words to be
entirely reliable, they should not proceed from a human source…If a
man has learnt it from some other man, he must have learned it from
some other person and so on..but the original source itself may be
unreliable. Therefore, they cannot be from a person…. Just as the
words of persons born blind cannot be a reliable source of knowledge
regarding shades of colour.
For example, doing a yagna X begets heaven. How can this be verified by ordinary humans or senses? Therefore, one has to rely on apaurusheya shabda pramana to realize this.
There can be two epistemological approaches. One is to say that all utterances are false unless proved otherwise. The other approach is to say that all utterances are true unless proved otherwise. The Mimamsas take the latter approach. Most scientific laws operate this way. It is your job to disprove that yagna X begets swarga. As you can not do that, it is true till falsified. As swarga etc are matters that cannot be perceived by senses, they should have emanated from impersonal source and uncorrupted by the human mind.