In my last game I was working for a space race on emperor. (something different to the normal domination)
I picked up the Incas as a peace time vassal.
By the late 1800s he was getting very close to winning a cultural victory.
Dozens of spies later, causing city revolts and demanding all of his resources resulting in all of his cities dropping below 10pop.
In the end he broke away and I got to run through and raze the cities in question.

So the questions are,
Can a vassal win the game?
Or does the master get credit for the win?
And is there a better way to cripple a vassal's ability to produce culture than what I did?
Or even a way to declare war on my own vassals?

Can a vassal win the game?
yes

Or does the master get credit for the win?
no

And is there a better way to cripple a vassal's ability to produce culture than what I did?
Spies are definitly a good idea, don't know about others except razing

Or even a way to declare war on my own vassals?
If the capitulated, you can't; otherwise, I don't know.

If a vassal doesn't have a tech needed to build any specific part of the spaceship, you can order him to research something else instead, thus preventing them from getting this victory type (although I guess they could still trade for the other techs, unless you disallow that). I'm not sure there is a way to stop a cultural victory (I never allow those anyway. For me only conquest is a valid victory condition, although I leave diplomatic on too because I like owning the AP).