I walked into the Phuket provincial health department in Phuket town when they were doing walk ins. Not sure if they are still doing them though. But worth visiting to check. I turned up at 0830 and they opened at 0900. After the first jab they gave me a slip to take to vachira for my second jab. Also vachira we’re doing walk ins. Although very busy and and a little confusing. Again get there early maybe 0800, as they issue numbers. I think only 200 for the morning I was there.
if you go to your line app and use the link below
https://liff.line.me/1645278921-kWRPP32q/?accountId=phuketwin
when you go to phuketwin there are some options at the top of the page. All in Thai though so you will need a friendly Thai to translate, unless you are proficient.