There are different type of vegetarians, such as Lacto ovo vegetarian, Lacto vegatarian, and Vegan. Some cultures eat insects, which may violate the naming of these rules. I'm curious if there is a name for vegetarians that eat insects too.
Logic suggests entomotarian by analogy with pescatarian and entomo- as a root for words related to insects. Specifically entomophagy is the practice of eating insects.
I'm not alone in suggesting this: a blog called the culinary linguist users of at least in passing, and there are other hits on Google