You can't use a cloud server. Any computer on the school's network can be set up to function as the server as long as it meets the minimum system requirements for FileMaker Pro, but the faster the system the better performance you can get.
There are also FileMaker hosting companies that can host your database for you in exchange for a monthly fee.
One or more databases hosted by Filemaker Pro or Filemaker Advanced allows 9 different copies of Filemaker access it (them) at one time.
5 concurrent (at the same time) can access the database(s) through browsers with IWP.
IWP requires the computers to be on the same network, OR the port (usually 591) can be set to be forwarded to the hosting computer.
http://portforward.com gives the instructions for doing the port forwarding on many router models.
Two copies of Filemaker will not change the 9 FMP user limit or the 5 user IWP limit for accessing the database(s).
You don't clarify whether you are looking to access the file from outside your network (such as from home) or whether you are trying to minimise the number of FMP licenses you need.