Getting Married in Phuket is possible and this is the process for a legal wedding in Phuket & Thailand.
To get married legally in Thailand you must visit your Embassy in Bangkok about 10 working days before your wedding date (+20 days for Philippines, +3 months France, so all embassies have different criteria ). It used to be quicker but the Thai Foreign Affairs have removed the express Service.
Each Embassy have their own country specific regulations and document requirements for the Affidavit. We do advise you to call them first on what these requirements are, as the rule and regulations are changing monthly.
The embassy document stating you are free to marry (Affidavit) is then translated into Thai (we can recommend a translation company) (at this stage you are now free to leave Bangkok) and the translated affidavit is passed to the Thai Foreign Affairs office in Bangkok which our office in Bangkok completes.
The normal paperwork required in Bangkok is passports, National Id numbers etc and divorce papers if you have been married before. You must make an appointment online and normally most embassies allows appointments to be booked up to 1 month in advance.
This paperwork from Bangkok is posted to contact in Phuket i.e us. The paperwork is further processed in Phuket local district (Kamala we use Kathu district ) about 2 days (6-8 days have passed since you started in Bangkok) and when signed you are legally married in Thailand. You can sign at the government office or we can arrange for an official to come to the wedding ceremony.
The paperwork, when translated back into English? is then recognized by your government when you submit it to your local office at home. (if you have left Thailand, we will post the marriage paperwork to you)
I hope this helps and from experience, most couples leave the embassy appointment too late hence our advice above.