It can be both a noun or an adjective. Veganism is listed as a noun, but we use the term vegan to describe ourselves, but as another person on this board states- "we" are not our actions. More I think about that the more it makes sense!
It's used as an adjective to describe food and products.
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It could probably be an adverb too (although no example immediately pops to mind, so maybe not).
EDIT: OK, here's vegetarian as an adverb: vegetarian specialty product. vegetarian functions as an adverb modifying the adjective specialty. You could also read the phrase as vegetarian and specialty both being adjectives modifying product, rather than vegetarian modifying specialty.
Dave in MPLS / DISCLAIMER: I am not an actual rooster.
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I believe that the main purpose of "a"s and "the"s in English is to make life of a non-native speaker miserable and not for distinction between adjectives and nouns. With this in mind, everything depends largely on context. If you say "She's a vegetarian girl", it will be an adjective (as name may suggest, used when added to some other word), but in "She's a vegetarian" it will be a noun (otherwise it would feel unfinished, like "she's nice", but "she's a nice ..."). Without "a" it can be an adjective once again (you'll probably use it when answering questions, like "Is it a vegan or vegetarian dish? It's vegetarian"). In a genitive context "Hummus is one of vegetarian's most favourite dishes" (I don't know if this particular sentence is grammatically correct however ) it will be a noun once again. Ok, sorry once again, everything I wrote lately is due to the lack of oxygen.