"Phonotactics" should probably be divided into syllable structure (and contact), plus syllable-count, since you would get different results if a language was Hawaiian with only 2 and 3 syllable words, versus Hawaiian with 2-12 syllable words. Looking at "words" is also somewhat problematic, since words may have a range of inflections, and so-called phonotactic restrictions might actually be broken by inflectional endings (as is the case with English monoconsonantal inflections). It would probably be more productive to look at phoneme combinatorics of stems rather than words, though "stem" is not a totally trouble-free concept.
This hasn't been done systematically (so it isn't a named sub-field), but it might yield interesting results. There certainly has been a history of making claims about gaps in the lexicon, which rise to the level of being a possibly publishable result in case someone discovers an interesting pattern. The main problem is locating comprehensive electronic dictionaries with interpretable phonetic data in a decent range of languages. The runner-up impediment is lack of knowledge of combinatoric possibilities in an interesting range of languages.
Theoretically, you could do this right now, using the CMU English dictionary, as long as you can compute all of the possible 1-syllable, 2-syllable up to ??12-syllable words of English, and see how many and what type are actually attested in English. I don't know of any large-inventory (Taa) or small-inventory (Austronesian) languages for which there are comprehensive computer-friendly dictionaries.