Literally the least knowledgable about this but I imagine you just go to a random clinic/hospital offering walk-ins and ask for one. Google says they're "quite cheap" (<1500 baht?).
Sorry, could you expand on this comment? I was able to book a flight to Chiang Mai last week; the flight stops in BKK and then goes to Chiang Mai via Thai Airways. Now I'm questioning if I need to do a stopover in Bangkok to do the 1 Day Quarantine, or if I'll be able to do that in Chiang Mai (because on the AQ directory website, they do have a Chiang Mai AQ hotel pending...)