Thanks for posting.
Hopefully, we get some feedback from posters using FileMaker in a similar way. In the meantime, a great place to start looking at examples of similar solutions would be the FileMaker and K-12 Education section of the FileMaker website. You can access this section here:
Hope this helps.
It can depend on a lot of factors but the first one that comes to mind is the number of students who will be trying to register at the same time. What size is your student body? Another issue is the level of processing required. Class scheduling can be simple or very complex. How many classes are being offered, # of subjects, do they need the student to have pre-qualification etc. This can also greatly affect the number of concurent transactions. Unfortunately FMS 10 still can't be load balanced easily and you are limited to a maximum of 3 boxes (data server; web publishing; web server) - that said with today's affordable octo cores and fast RAIDs they can go a long way. The throughput of your internet connection can also affect the number of connections.
That said, unless you're a large non profit organization, FMS should be able to do the job.