Set up a field that is empty when the record is new and not yet submitted and which get's a value when it is. This could be set up any number of ways via script or auto-enter fields. (A field that auto-enters a serial number "on commit" comes to mind...)
Then set up custom access privileges on this table with a lock expression that only allows editing the record if this particular field is empty.
See "Editing record access privileges" in FileMaker Help and check out this particular sub section: "Entering a formula for limiting access on a record-by-record basis" for a description of how to set this up.
Worked great, thanks for the quick reply!