Vampires' bodies are dead. They are people who died whose corpses were reanimated by evil forces and who thus thirsted for the blood of the living. Originally, hundreds of years ago. That's it. You can destroy their bodies, but not their spirits. Eastern Europe has dozens of variations of vampiric lore, the most famous of course being Vlad Dracul.
My point is, just because their bodies are hosting their evil spirits doesn't mean the body still lives or is affected by disease. The only reason you can even kill one is because whoever invented the tale centuries or millenia ago needed closure. No one wants a blood thirsty demon running around that is unstoppable. Thus stakes, garlic, sunlight, etc. It made the uneducated and illiterate people who fell for the superstition feel safe, like there was something they could do to protect themselves.
But logically, disease wouldn't do anything. Vampires operate outside the bounds of normal anatomy and mortality.