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

I often search for the etymology of words. I used to overthink it when trying to find the right search terms but eventually realised that "[word] etymology" gives pretty much perfect results every time ('vaccine etymology' gave the answer on Google and DuckDuckGo without even needing to click any links).

Just mentioning this for other people who like looking up words. And yes it's obvious... unless you're one of those people who sometimes overthinks search terms.

At least wikipedia seems to partially agree (on the cow part), with citations I couldn't verify:

https://en.wikipedia.org/wiki/Vaccinia#History

https://en.wikipedia.org/wiki/Cowpox