My clients connect to our server via Instant web publishing. I was wondering if I could have a custom menu viewable to them that had a print button that told the server to print to the default printer?

IWP doesn't do custom menus and FM Server doesn't print. You'd have to use a dedicated FileMaker Pro client machine to do the printing, and create scripts to somehow queue up the print jobs.

Or they could use the Print command from their browser.

True, but even so that can only get you so far (e.g., IWP limits the number of records in a list). But I was picturing the printer in the office and the user at a remote site.