The general consensus is that, phonetically, early PIE had a low vowel of some sort (which could broadly be called
[a]), but phonemically, it did not have a phoneme
/a/ (or it had a marginal one at best).
The reason for assuming a phonetic
[a] is that various words have a consistent low vowel across several daughter languages, like in "before, fore, front" (Latin ante, Greek anti, Sanskrit anti).
The reason for not assuming a phonemic
/a/ is that, for the most part, it doesn't seem to be necessary. When we see those consistent low vowels, the laryngeal theory explains them as an underlying
/e/ adjacent to
/h₂/. And indeed, Hittite shows a direct reflex of
/h₂/ in the "front" word: ḫant- "face, fore". (Reconstructed PIE phonemes are generally written like *this rather than
/this/, and that's what I'll use for the rest of the answer.)
Why not assume a phonemic
/a/? Because this "laryngeal-coloring" theory very nicely explains certain inconsistencies in ablaut (the way the main vowel within a PIE root changes in different forms). For most roots, there are three common ablaut grades: ∅, e, and o. But certain roots seem to replace the e-grade with something else: sometimes ō, sometimes ā, sometimes ē. And these same roots do something weird with the ∅-grade, which varies by language; in Greek, the ∅-grade of those roots show o, a, and e, respectively (Greek didōmi "give", dotos "given"; histāmi "stand", statos "stood"; tithēmi "place", thetos "placed"); in Latin, the ∅-grade is always a (datus "given", status "stood", factus "done").
If we assume laryngeals, these inconsistencies completely disappear. Those roots are now reconstructed as *deh₃, *steh₂, *dheh₁; the weird long vowels are the result of a laryngeal after *e; the weird short vowels are the result of a laryngeal on its own without a vowel next to it. And the ablaut is back to the classic ∅ vs e vs o, same as any other root.
And once we've added the laryngeals to the theory, well, it turns out they can explain basically all instances of phonemic *a. Some linguists still reconstruct *a as a marginal phoneme in words like *aból "apple", while others explain even these instances away as *h₂eból. At this point, adding *a to the theory as a full-on phoneme makes it more complex without adding much explanatory power, so Occam's razor suggests we should leave it out.