Does anyone know the origin of the theory that the vowel /a/ is inherent in all consonant letters in the Indic scripts? I see it repeated all over the place without any reference to where it comes from.

It works to some extent as a handy way of describing how the system works for Sanskrit, but from a phonological point of view it's implausible as a description of what's really going on, not only for Sanskrit alone but also because of the variety of vowels that can be "inherent" depending on the script. And as Lloyd Anderson points out in his review of _The World's Writing Systems_, it doesn't work well for describing how the writing system of modern Hindi (and I can add Gujarati) works. And for the Sumatran scripts (Batak and Lampung in particular) where vowel marks are shifted from the syllable initial consonant to the coda consonant that bears the virama, the "inherent a" theory fails miserably and forces you to use all sorts of workarounds to describe what's happening.

Any ideas about where this theory comes from?

Sure it does. Consonants always have the vowel a in Hindi, except when other vowel diacritics are added, or the virama is used (most often expressed with the use of conjuncts). The exception is at the end of words where the final -a is usually not pronounced, but it was in Sanskrit.

This isn't true, however for some languages like Bengali, where the inherent vowel is o or ô. But this is due to a phonological shift, not the inherent design of the script.

OK, the theory "works" when viewed in isolation, and I didn't point that out. It "works just fine" sort of like relying on duct tape to do repairs does. From a metatheoretical standpoint, the theory is suspicious because of its unnecessary complexity. It's just one way of describing what's going on, and one that obviously comes from a time before the insights that became available with nonlinear phonological theory.

The "inherent vowel" theory (no matter what the particular vowel may be) posits a general rule inserting a specific vowel to be read with each consonant. (This is the economical version: an even more complex version would list each consonant letter with the same vowel, thereby missing an obvious generalisation.) After having inserted this vowel by rule, the theory then requires a set of rules to "delete" it, and has to specify under what circumstances this "deletion" takes place. Already for Sanskrit, this fails in the face of the principle of economy (aka Occam's razor).

Compare this to a theory in which: (1) the only inherent and invariant content of any consonant letter is the consonant phoneme itself, (2) the virama simply specifies that there is no syllable rhyme (in practice, a vowel position) following a consonant, (3) vowel matras merely add their vowel to the consonant (in syllable rhyme position), and (4) if nothing else is specified, regular phonological principles of syllabification (and not special orthographic rules) determine where the phonological (not orthographic) default vowel will or will not be inserted.

For Hindi, you have to posit extra rules that count syllables to decide where the "inherent vowel" is "deleted" (after being inserted by the general "insert 'inherent' vowel" rule) rather than simply let the native reader's knowledge of phonology and syllable structure do the work. (See Lloyd Anderson's review of The World's Writing Systems for a critique of the "inherent vowel" theory and its application to Sanskrit and Hindi.) And this is just for the simple cases. For other cases involving affixation, where syllabification principles and last-resort default vowel insertion work just fine, you have to posit workarounds to get the ever creakier "inherent vowel" edifice to work. This is what was done in Kachru, Kachru and Sridhar's (2008) Language in South Asia, where they admit that their syllable counting analysis doesn't work beyond the simplest cases and refer the reader to Ohala's 1983 analysis of Hindi phonology, critiqued by Norval Smith in 1984 for its unsatisfactory nature:

"The account of ə-deletion is perhaps the least satisfactory part of the book and points up the need for a non-linear analysis, using some form of sonority hierarchy, to predict the consonant cluster environments in which ə may and may not be deleted.
The overall impression received on reading this monograph is one of irritated disappointment tempered by considerable admiration (...)".

The theory just adds unnecessary complexity that doesn't stand in the face of our current understanding of a native speaker-reader's phonological knowledge.

Without going into details here, it gets even more complicated for Batak and Lampung in Sumatra, which have a special spelling rule that shifts vowel marks from the syllable onset consonant letter onto the consonant letter that is followed by the virama.

In any case, all I wanted to know was if anyone had any idea where the outdated and unsatisfactory "inherent vowel" theory came from originally. I've managed to trace mentions of the idea to the mid-to early 1800s but the farther back you go, the vaguer people get about where their ideas come from, our modern citation conventions not yet being in vogue. On the other hand, I have a fairly good idea of where the latest illustration of this theory comes from (Ohala 1983), and where the critiques and alternative viewpoints are to be found (Smith and Anderson). I just want to get as accurate as possible in my citations while adding a more complete and balanced discussion of abugidas to the Wikipedia article(s). (As well as for a possible future journal paper on the role of phonological syllabification in script typology.)

OK, I went at the problem from a relatively theoretical viewpoint in my last post. Here's a layman's version (for any abugida). Just two common-sense instructions, and one rule:

Pronounce the consonants you see written. (D-uh!)
Pronounce the vowels you see written with them. (D-uh!) (Extra instruction: after the consonant, no matter whether the vowel is written above, below, before, or after the consonant.)

Special rule: If a consonant is a reduced form or has a virama, it can't have a vowel after it.

The rest comes from your knowledge of the language, not from spelling rules:
You already intuitively know when it sounds "right" or "wrong" to have two or three consonants together: where it would sound wrong, you're going to want to pronounce a vowel there. You tend to do that in loanwords from other languages anyway.
You already know that vowel is X (/a/=[ə] for Hindi or Gujarati). (It's the one you pronounce automatically if you read a consonant on its own out loud.)

No need to say "it's always there except when (a, b, c... etc.)."