Vaccine comes from the latin "vaccinus", a derivative of "vacca" (cow) and means "related to cow" or in this case more like "of cow origin". For example in italian "latte vaccino" means cow milk. So there is no -cine part, only a "-inus" suffix that simply means "related to" or "coming from". You can see that in a lot of words of latin origin. In fact the term "latin" itself comes from "latinus" which means "related to Latium" or "inhabitant of Latium" (a region in central Italy)

Unrelated, but as for the origin of the latin term "vacca", it may come from vedic, with the meaning of "the animal that moos"

Edit: made my answer more to the point