I use pcloud.com for a similar problem.

They have custom HTTP JSON RPC protocol, it uses POST and GET verbs. Their web site calls that one. They also have more performant TCP binary endpoints for use by native clients, with the same payload packed into custom binary format.

For users, it's a web app with download and upload links.

How would that look for assignment submissions? Something like WebDAV?

Just a page with a form you can upload files through. You can setup something like that e.g. with Owncloud/Nextcloud. And dedicated education tools also often have it.

For normal document-sized files, email is fine though too IMHO, for university stuff I never really cared which way was used.

NextCloud is quite a dependency for a simple file upload.

Sure, if you just install it for that. But it's a reasonably common service (e.g. around here, many universities have it for their employees and students) with that feature.