"Can", in this case, actually has 2 meanings. First, will the ROM work on that phone? If there's no version of CM for it (and that means for the CDMA version of the G, not just for the G), there's no reason to ask the second question, which is the one you asked. There's no reason Verizon should shut your service if you're using a different ROM, as long as the ROM doesn't try to do things Verizon doesn't allow - like not charging the minutes you use. (And Cyanogen doesn't do things like that - they're now a respected part of the cellphone industry.)
But the best place to ask about CM ROMs is on the CM website. They know their ROM a lot better than anyone else, and would have more experience with how Verizon (or any carrier) would handle use of it.