Traditionally, in Indo-European reconstructions we reconstruct *e and *o as the only two vowels, and thus, no such thing as a vowel *a. We generally explain *a's away with laryngeal colouring (Notable *h2, but some people also come up with a *h4).
Some people argue that the vowel system that is formed by an absence of an /a/ is highly unnatural since the vowel system would look something like this:
This system is considered unrealistic because it is very sub-ideal. A vowel system tried to have the maximum amount of contrast, so the most maximal distances from each other. Obviously *e and *o, which could also be [e] and [o] rather than how I presented them as Open-Mid vowels, are not the furthest possible removed from i and u.
In a system as presented above, on an exam question 'fill in the missing vowel to make a natural vowel system' not a single person would hesitate to fill in the [a] at Open Front/Open Central.
So, to make this vowel system look more natural I came up with the theory that maybe the vowels are rounded back and front vowels in the 'Open' position creating a vowel system something like this:
The system would then look something like this. I proposed that maybe the reaches of these vowels were quite extreme in certain contexts.
These reaches of the vowels I've done with the shades on this diagram.
It's similar to how the Arabic /a/ tends to sound more like [æ] especially when not surrounded by any laryngeals. While it takes on the sound of [a] when it is surrounded by laryngeals or an /r/ (Which in Semitic languages is similar to a laryngeal). This Would actually perfectly fit our theory of *h2 being a pharyngeal colouring our '*e' (or [a/æ]) into an *a-like sound.
The reach of *o is quite a logical conclusion since the Indo-European *o tends to become a in a lot of languages (Germanic, Hittite, Balto-Slavic...). This vowel 'box' would also be a very nice explanation why *e and *o merged into /a/ in Sanskrit, which is with the above theory (the i-u-e-o one) very hard to explain.
But these theories only make sense if you assume that there in fact is no phoneme /a/. And although this phoneme is so incredibly rare it's hard to believe it exists, denying its existence is very unscientific, but is still done by the majority of the current Indo-Europeanists.
But there's reasons to assume that the PIE *a is not originally from Indo-European, for example, it doesn't partake in any ablaut paradigms like you do see with *e/*o. And besides that, there's just an extreme scarcity of the phoneme all together.
But like I said, it's quite unlikely to assume there is no *a, or never was. Which brings us to assuming that maybe originally the *a was there, and due to certain developments it just managed to get lost in many many contexts.
Now we get to a rather controversial part of my story. Some linguists assume that PIE was originally related the Uralic languages, creating a Indo-Uralic so to speak*. These people assume that the PIE branch of Indo-Uralic moved towards the Caucasus, and coming in contact with Caucasian languages, which tend to have big consonant clusters, and very small vowel systems (2 vowels /a/ and /ə/ if I've been informed correctly) PIE underwent a huge syncope of vowels, except under the accent. We still see strong correlations between the place of the accent, and the appearance of a vowel in Indo-European.
Now let's assume that maybe, just maybe *a was a vowel that could only appear OUTSIDE of the accent, that would explain the rather abrupt loss of the vowel in Indo-European. /a/ appearing as a non-accented vowel isn't without precedent either (or more accurately postcedent). In Russian the phoneme /o/ only contrasts with /a/ on the accent syllable, otherwise the two merge.
So *a sort of performed a function similar to a schwa (ə) in many languages. This is not unthinkable, if we think back of what is a 'natural' vowel system. If a language has a schwa, but no a, this schwa will automatically try to extend it's quality towards this a.
Now what exactly the conditions were to retain this *a in some positions, I don't know, but it's in no way unthinkable that a far-reaching syncope would not be able to affect ALL cases of the *a. This would explain both the scarcity and the existence of the vowel.
Just for completeness let me give you a little diagram of what this pre-syncopatic vowel sytem:
* Let me especially refer you to Kortlandt's work which is to be found here. Pay special attention to this very interesting, but not really related article on the Indo-Uralic verb