They can get one from a nearby Consulate (Vientiane maybe, or your home country). Then you can add them to your Extension. If they are on tourist visas you can change it to 'O' visa at Chaengwattana Immigration then add them to your Extension.
I got my son's UK passport in 2015. His Thai birth certificate was translated by a translation company in Ploen Chit. There was no need for verification from the Thai MFA. I doubt it's changed since then.