If they can only approve "their" invoices, will there ever be a case where they might need to edit any other fields on an invoice that isn't "theirs"?
If you can make this work by locking them out of editing any part of the record when it isn't theirs, you can set this up with Privilege Set options specified for their account.
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.
If they need to be able to edit other fields in the record, just not this approval field for records that aren't theirs. Let me know as it's still possible but the options for doing so are more complicated.
This wouldn't necessarily be on a "record-by-record" basis. For example, a single record represents a single sale. Within that sale, the legal department, the accounting department, and the technical area must all approve the invoicing and the order. Therefore, I'd like each area to only be able to select the radio button specified for them, and not be able to edit/delete other radio buttons.
This is to ensure that certain areas are not approving other people's radio buttons to "accelerate" the process.
I'd set up a table of related records in a portal for the approvals. One record for each approving entity. Then the record level access control method that I described earlier can be used to limit their ability to edit records in this related table to only the ones you want them to be able to edit.