It can function, the easiest way to set it up would be for the iPads to generate attendance files that get imported to the desktop version later. Make sure to use UUIDs as primary keys to make sure records are completely unique.
For getting data out to the ipads (EG attendance) you could use the same sort of export/import package style format.
The benefit to using server would be the instant accessibility of all attendance data to all iPads, but it sounds like the iPads are entry only, and only the desktop would need the reporting access.
You could also use peer to peer networking to connect the iPads to FileMaker Pro. This should work ok since there will not be a much interacting with the data base. If all you want is 4 teacher to enter attendance, FileMaker Server might be a bit of overkill.
Hosting is cheap and easy.
You can find it for $15 bucks a month! It'll pay for itself every month in ease of use.
I must chime in supporting David's statement. Best Practice hosting is with FM Server (not peer-to-peer from Pro). FM Server offers the best protection for the data and assures timeliness and up-to-date accuracy which cannot be assumed with periodic imports instead of direct Server client. Peer-to-peer is notorious for data loss and file corruption in ways that proper FM Server hosting avoids.
Could you suggest a few companies that provide this type of service?
To get you started to find what's right. Although I still think you don't need to host it if all you are doing is data collection.
You should be able to collect data and send the files of collected data back to a central location. There is manual work involved, but for four teachers it seems fairly easy.
If you need up to the minute / realtime reporting on attendance, then yes, hosting might be best.
Thanx for the links that is helpful.