The ideal solution would really be for the shared computers to be set up such that students had accounts with roaming profiles, such that they could just use Zotero and have their data there when they logged in. As an alternative, they could set up Zotero syncing when sitting down at a computer, let Zotero sync down their data (ideally with download files "as needed" selected to avoid downloading all files) and then unlink the account and choose to delete data before logging off the computer. Unless they were working with very large libraries, either of these approaches would allow them to use the normal Zotero app on a shared computer without unreasonable delays when starting work.

If those options aren't realistic, note that they don't necessarily need to manually insert items when using the web library. The new beta web library — accessible via a link at the top of the web library — includes an Add by Identifier button that works similar to Add Item by Identifier in Zotero, except that it also can take URLs (similar to ZoteroBib). They can also use the save page, which is equivalent.

It's also possible to save directly to the online library with the Zotero Connector, which would allow saving from gated databases and include accessible PDFs. When trying to save using the Zotero Connector without Zotero being open, it will prompt them to whether to save to the online library, and then not ask again unless Zotero becomes available. This will sometimes provide better results, but the downside is that they would have to be careful to log out ("Clear Credentials") from the Zotero Connector preferences when leaving a computer if the shared computers don't clear themselves on user logout.

Of course, while the beta web library improves the experience quite a lot, it's still not equivalent to the Zotero app — e.g., no word processor plugins, file management, advanced search, etc.