There isn't a clear cut answer to your question as it is a cost vs. benefits type of decision.
The "cheap" way to go is to install FileMaker Pro on their server. Then additional copies of FileMaker Pro would be installed on each workstation and they'd use "open remote" to access the database. This works for up to 9 or 10 clients accessing the DB at the same time when hosted by FileMaker Pro.
It also possible to use Instant Web Publishing (IWP) and then up to 5 simultaneous users can access the database from their web browsers, but this imposes significant limits on your interface design (Script triggers don't work among many other limitations), so this may not be practical for your current design.
Assuming that their server meets the hardware requirements for FileMaker server, you can install server on that machine instead of FileMaker. This provides better protection of your database, faster performance, automated backups and the ability to set up scripts to be run from the server on a schedule--all very useful features, but they come at a significant added cost. And if you use IWP, you have to install FileMaker Server Advanced.
In addition to all that, I strongly recommend a single copy of FileMaker Advanced for you to use to make design changes to your database. The added developer features in that version can save you a great deal of time and offer design options not possible with Pro alone.