I am assuming when you say they are connected "wirelessly to the internet" that a device in the school that is what gives them internet?
Budget is how much? How much space do you need?
There are a bunch of ways you could do this but I think it all boils down to 2: build and fiddle with a server, windows or linux as the OS OR get a NAS box with a few bays/drives.
If you went windows as the OS the computer is going to have to be much beefier(aka more expensive) than for a linux distro but the windows would be a bit more familiar and easier to setup. Oh and then there is the pesky licences for windows as well.
Where as a NAS and I am talking more of a pre-built one, would be plug, configure users/security through a GUI, and pretty much done. You could build a NAS box, which would be less expensive, much more flexible but again there is the time factor of putting it together, fiddling with it.
In the end I would look at doing a NAS. I'd go pre-built NAS, they aren't that expensive and are fairly straight forward in the setup. But if you have no budget building one now a days isn't that difficult and once it's built the setup is almost the same as a pre-built.