What's the easiest way to get a references section into a single citation format if they vary to begin with?
Doing this in a reference management software (I use Zotero, which you mention, but there are others) is absolutely the way to go. Of course, setting up the reference database for one paper may take the same amount of time as manually reformatting, and learning how to use it in the process may take longer. The advantage is in reusability. For any single paper the up front cost may not be worth it, but the time savings on paper number 2 and beyond is incredible.

Unfortunately you won't just be able to paste in the formatted citations and have Zotero recognize them (so far as I know). Zotero does have a way to add items by an identifier, so if your coauthors included DOIs, adding the references should move quickly. Additionally, if your coauthors used any reference management software themselves, you can ask them to export it to a common format (BibTeX is pretty commonly supported) and import those files.

If worse comes to worse, searching for each reference title in Google Scholar and using Zotero's automatic ingestion should also be pretty quick.

Zotero's documentation on these various ways to create items:

https://www.zotero.org/support/getting_stuff_into_your_library