You’ll need to apply for an import license through the Thai Veterinary Authority (
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(this link is for Bangkok, there’s other email address when flying into Phuket). They will also tell you what their requirements are in regards to vaccinations, passport, health certificate from the origin country (cannot be older than 10 days, most airlines will also require this), etc.
Check the requirements of the airline you’re flying with.
Check what is needed to bring your dog back to the US. Probably you’ll need to do a titer test, best to do this before you leave for Thailand.
It’s all a bit of paperwork, but I always find the people at the Thai veterinary authority very helpful (I’ve traveled back and forth with my cat a few times, to EU)