I found someone that I think might be a vampire, but he's married and has a young child. His wife is somewhat suspicious too. I'm thinking of just executing the three of them. Can a vampire marry a non-vampire? Would their child be vampire?
No, it is not, because vampires are supposed to be sterile. (Note however that it has been reported this feature isn't working right.) So if your putative vampire has a wife and kid, and they both acknowledge him as the father, either he isn't really a vampire, or it's from his previous life as a mortal. Note that mutual agreement about relationships is important. Vampires will falsely claim to have relatives, even spouses and children, in distant lands, even if no such dwarves exist; only relatives also at the fortress are safe.
In any case, the wife and child are probably perfectly normal dwarves, unless they have independently become vampires themselves (which is vanishingly rare). Vampirism cannot be transmitted genetically for the simple reason that vampires cannot have children (barring bugs as above). In this case, since the child is young, I would probably guess that the father isn't actually a vampire; what has he done that inspires you to think he is?
Note however than executing the father will cause an impressive bad thought for his wife and child. If you're not prepared to handle this, and they're new immigrants so they don't have other friends in the fortress, you might as well execute all three.
vampires themselves are sterile, but the guy might have contracted vampirehood after getting the child