That is a strange one. Pressing tab should produce the same behavior whether the file is opened with FileMaker Pro or the runtime executable.
Could it have something to do with custom menu sets? I ask because I have had copy / paste functionality break when it wasn't included in the menu originally. The setup I have is fairly limited:
I doubt it.
As a test, launch FileMaker Advanced and open the actual run time copy of your database file using the same password that your run time users will use. Test it with FileMaker advanced and see if the tab key moves you from object to object in the layout's specified tab order.
Details to check:
Use set Tab order to confirm that the layout does have a tab order
check to make sure that Behavior settings in the inspector specify tab as a key to exit the field on each field on your layout.
Look for scripts triggered by OnObjectExit or OnObjectSave that might change what object next ends up with the focus...
"Could it have something to do with custom menu sets?"
I don't think so; maybe that you have some OnObject Exit trigger ?