Check to see if iCloud is enabled - especially iCloud backup and iCloud Drive. If it is, then that is the method the school has chosen. They may have also given your child access to a school website or app where they can save their work.
Generally, schools do give students a method for saving their work to the cloud. As has already been said, you should consult your child's school to learn what method has been provided.
When schools issue iPads to students they take steps to prevent the students (and their parents) from changing the configuration of the iPad. iTunes is blocked, as iTunes is one of the ways a "normal" iPad would be erased/reconfigured.
I couldn't find any iCloud/iCloud drive on the iPad, but I'll take another look.
The kids have been instructed to use a specific app for uploading their assignments and reading what to do etc. and that works fine, but this doesn't help work that is being worked on, thus not already uploaded yet.
So I was hoping for a way to back up the iPad files locally for that.
Yes, indeed -I had forgotten about iTunes being able to erase an iPad, which is probably the reason as you say they've set up restrictions.
I'm going to ask the school for more info about all of this as it's obviously not self-explanetory.
