They'd stay the same. As we've seen with Edo Tensei, physical changes aren't really permanent. As seen with Shin, you can even completely destroy their bodies and they'll simply reform to the way they were.
Assuming we're dealing with the typical "Vampire bites a person and turns them" type scenario, then biting an Edo Tensei would be like biting a corpse, they might be able to get some blood out of it, but the corpse isn't going to change in nature.
Alternatively, as we saw with Hanzo, poisons can still affect Edo Tensei bodies, so if we're approaching vampirism as an infection type of thing, like Daybreakers, then it might, if only temporarily, turn the Edo into a vampire.
Well... basically are you asking if an edo tensei can become a vampire?
If so, as far as I remember the majority of the myths on vampires says that to turn someone into one of his kind a vampire must first of all drain him of his blood, leaving him in a state of near-death, and then feeding him the vampire's blood.
Given that Edo Tensei should not have blood, and that they already are past death, I guess that nothing would change