How are you unpausing and telling FileMaker to continue? You can always capture keystrokes of any input and evaluate them to decide what to do including looking for ASCII 10 or 13 (or 9 for tab) depending on what you want to do. Is the RFID sending multiple records/fields? If so, push them into a global field and then parse them out with a script into the appropriate fields.
The field that is getting the RFID info - is it set to go to the next object with the Enter and Return key? Check on the Inspector.
I've worked with a lot of scanners (though not an RFID scanner). Most of the scanners I've worked with are basically seen by the computer as a "very fast keyboard" when it comes to the data being entered in the field. I've done exactly what you are trying to do, but with barcode scanners. Everytime it seems like it is a bit of trial and error to get the scanner setup correctly. Typically I like to scan into a text editor during this setup time so I can easily see the exact characters that are being sent.
I'd start by scanning into a text editor and making sure the scanner is in fact passing through a return or enter character. Then once that is confirmed move on to troubleshooting FileMaker and making sure the field is setup as David suggests, or capturing the characters if needed as Taylor suggests.