Usage of "kick in"

Legal? At least in the UK the law doesn't normally control the language we use. But, yes, kick in is a phrasal verb meaning 'come into effect' or 'come into operation'. You don't need to worry about it being followed by another in. In fact, in your example it's necessary.

You could always say

The rollover will kick in at the next billing cycle.

or

The rollover will kick in on the next billing cycle.

EDIT: (to answer the "is in required" part)

Either would avoid the repeating "in", which I think is helpful, but not absolutely required. Without it, the phrase "the next billing cycle" is adverbial, like saying "next month", so it would be like saying...

I am going to pay my bill on time (the) next billing cycle.

...which sounds awkward to me, because a billing cycle is another kind of beast that has dates associated with it, but not precisely the same thing as a time span like "next month".