One of the lessons that was pounded into me while trying to learn Russian and German (as a native English speaker) is that prepositions are a royal pain in the ass. In this case, I think that you would be understood if you said either "addition on" or "addition in" $V$.
That being said, there are subtle reasons that "on" might be slightly preferable. One way of thinking of binary relations (addition, multiplication, etc) is as functions with a domain consisting of the two-fold Cartesian product of the set. That is, addition is a function
$$ + : V\times V \to V. $$
Addition can be defined on this Cartesian product, thus as a shorthand, it makes sense to say "addition on $V$."
Moreover, as others have pointed out in the comments, we often think of groups acting on other sets. In a really abstract way, we can think of an underlying groug acting on $V$ to give vector addition. I wouldn't worry too much about such details—at least, not until you have taken a few courses in modern or abstract algebra—but you might eventually be able to make some sense of it.
All of that being said, if I really wanted to emphasize that a set is closed under an operation, I might be tempted to say that the operation is defined "in" a set.
Long story short: Either preposition is fine. There are subtle distinctions to the native ear. "On" is probably slightly preferable in most situations.