So, presumably, at some point during of after the fall of the Roman Empire, Latin became a dead language. Or, at least no longer used outside of the Church or science. When that happens to a language, does that stop the process of borrowing words?
Yes, borrowing still happens—in both directions!
While Latin is dead in that nobody speaks it as their first language, it's still used for official purposes by scientists and the Vatican. When they need a word for a new concept, they have to either create or borrow one, just like for any language. For example, "internet" in modern Latin is interrete (calqued from English), and the Pallas cat has the biological Latin name Otocolobus manul (from Mongolian manuul).
On the other hand, words are still borrowed from Latin for all sorts of purposes, despite its death. "Television", for example, comes from Ancient Greek tēlē + Latin visio, and was borrowed long after the fall of the Roman Empire.
The key is that "Or, at least no longer used outside of the Church or science."
If a language is actually dead—not being spoken or written by anyone—then there's no way for it to borrow words. However you choose to define what "language" means, there's no speech or text that will use the new words, no speaker who will ever think them…1
But medieval and post-medieval Latin was only dead as a vulgar language—as you said, it continued to be used in the Church, and in science (and in other areas, like international diplomacy and often even internal bureaucracy).
Theologians, scientists, engineers, etc. were obviously coming up with new ideas that hadn't occurred to the Romans. So, how did they express those ideas in the Romans' old language?
You can always use circumlocutions to avoid needing new words. But when you're talking about the same concept over and over, that gets annoying fast. So, what do you do? Invent a new word. You can try to invent it from first principles as you think a native Latin speaker would have, or you can calque a German or French word into Latin, or you can just borrow the word. All of these happened in Latin, just as they did in, say, French.
Borrowing into Latin isn't exactly the same as borrowing into French. Most borrowings into French are probably colloquial, and may spread to a sizable subset of the French-speaking community before anyone even stops and takes notice. Most borrowings into Latin, on the other hand, are done consciously, and explained directly in the paper that first borrows them, and spread by people being consciously convinced of the coinage's value. But they're both borrowing.
1. Even this isn't quite true. In theory, even a totally dead language could be resurrected. I don't mean just cases like Modern Hebrew (which is usually considered a different language from the older languages it was constructed from), but people who have taught themselves to, e.g., speak Old Franconian fluently, the same way others have taught themselves Klingon. They may have very small speech communities, and no L1 speakers—but if they're actually using the language at all, they could end up borrowing words. At that point, do we say that Old Franconian has borrowed a word from Modern Dutch, or do we say they're actually speaking a new language, Zombie Franconian? Or does it only count if they raise their kids speaking L1 Old Franconian, and their kids borrow words? I think it's really just a question of definitions, not facts. If cases actually arise that push the borders of our definitions, we'll come up with new definitions, or new words (and then they can borrow them into Zombie Franconian).