According the Bangkok Post article linked below, Expats can begin registering for their vaccine(s) on June 7th. The policies supporting vaccine distribution have been in a constant state-of-flux, so this start date may possibly change.

You'll need to Register at an Approved Hospital - so it would probably make sense to ask your local hospital about this in advance just to be prepared.

We already know that Moderna, which is the most effective vaccine at the moment (95% efficacy), won't be available in Thailand till around the fourth quarter...God willing, and the creek don't rise.

https://www.bangkokpost.com/thailand...rs-from-june-7