Yes, my labradoodle and I moved here from Kauai, Hawaii. The dog relocation services are ridiculously expensive from Hawaii, $6,000 and up! My dog flew on the same plane as I did, in cargo. It was harder leaving Kauai than coming into Thailand (rabies shots 6 months apart, notarized docs had to be fed ex’d by my vet, export permit, proper kennel, needed to travel when it’s under 85 degrees at all destinations so we landed BKK at 1 am, etc.) Cost about $2,600 from start to finish. PM me if u need more info.
Royal Thai Embassy page online will have the correct answer to your question. There are many types of Visas, it depends on how much you're willing to spend.