To exact revenge or vengeance?

There is no "definitive answer". Both forms exist...

And as you can see, until a generation ago most people would have said exact vengeance was the "correct" version. But (particularly in the US), the pendulum has swung massively the other way. Rather than toss a coin though, I suggest go with today's usage rather than stay stuck in the past.

I will just acknowledge StoneyB's point that strictly speaking you normally exact things like retribution (because exact means compel to be given, take forcibly). But he will have his revenge implies if you can have it, you can take/get it (from someone). By force, if need be.