How can I obtain a re-entry permit at the airport in Thailand?

You can get a re-entry permit at both DMK and BKK airports

In Suvarnabhumi the re-entry permit counter is in International Departure #2, it is after security screening but before you stamp out of the country. Look for the signs

In Don Mueang the re-entry counter is AFTER you stamp out of the country at passport control. Tell them when you stamp out you need a re-entry permit and they will show you where to go.