Since we haven't seen Vampire Animals in the Buffyverse yet, I'm going to assume they don't exist. But many other Vampire-stories do have them.
I guess that if you look at it from a biological point of view, you could say vampirism is like a virus. As far as I know, virusses always have a limited group of species that they can succesfully infect. Sure, some maybe able to target a lot of species, but I doubt there's one virus that could infect EVERY single species. Given that the vampire-virus can infect humans, the further a species is removed from the homo sapiens, the less likely it is that the vampire-virus could infect it. So vampire reptiles or fish not very likely. Squirrels, rats, could be fun, but prolly not. Dogs, cats, wolves, maybe, who knows. Vampire apes, wouldn't rule it out.