I've looked at a few of the signup websites for foreigners, but they all seem to require long-term visas.  Is there anywhere I can register since I arrived in 2020 as a visa exempt tourist?  I'm in Phuket and would like to get the vaccine here.

 

I'll be visiting my home country soon (where covid is even worse than in Thailand currently), but would prefer to get a vaccine before traveling internationally.

If you have been treated at one of the private hospitals, for example BHP, at any time, then you can register with them and most probably get your vaccine.  When they checked my passport, there was no check on my visa/extension, just my identity.  If you haven't been treated, maybe one way around it is to get an appointment for a medical check up and then request a vaccine. that may work but it will cost you.  

from many registration sites I have registered, only expatvax required copy of visa.

Try sinopharm, registering every wednesday morning and you would need travel for your shots to bangkok.

Be on a lookout mode for walk-ins on Phuket. Probably they will throw over 70%, to save sinking sandbox