I agree it’s probably worth asking a host what the animals eat, not only to see if it’s something you might object to, but also to see if it is something that may be very complex and hard to follow.
I don’t know what most hosts do, but it seems to me if you are applying to a sit and it is mentioned in the list that the host does not want meat brought into the home, you absolutely should ask the host what the pet eats if that is a dealbreaker for you. Going by vegans and vegetarians I know who have pets, and homes I’ve sat for, I’m going to guess that most vegetarians/vegans are fine with giving their pets (especially cats) meat (or at least kibble made with meat).
Many, but not all homeowners do describe what their pets eat in the listing, but if they don’t or it’s not detailed, this is absolutely something sitters should ask about.
My own listing for instance mentions that the cats eat canned food and need to be separated at meal time.) Some homeowners might for example feed their pets a raw diet, or a home cooked one, and some feeding routines can be overcomplicated and more than many sitters want to take on. My parents wound up being late in life dogowners when my brother’s work hours were too much and he gave the dog to my parents. My mother always fed the dog ground meat with vegetable table scraps. She always made his meals up fresh. I don’'t think I’d want to take that on as a sitter!