This is a fairly involved process. The first step is to have the marriage certificate authenticated by the state in which you were married. Then it needs to be authenticated by the US State Department. Then it needs to be authenticated by the Thai Embassy in Washington D.C. Once this has all been completed, your marriage certificate and all of the certifications above need to be "officially translated" and then be "legalized" at the Ministry of Foreign Affairs in Bangkok.
Once all of that has been completed you will be able to use your US marriage certificate as a legal document in Thailand. You and your wife can then register the marriage and your wife can request a change of marital status/name change or anything else that is needed at your local amphur.
When I did this a couple of years ago, I used an agent in the US for all of the certifications, then an agent in Bangkok for the translation and certification of the document at the Ministry of Foreign Affairs. I was also required to get a certification of my passport at the American Embassy before the amphur would issue the Khor Ror 22.
Total time for the US side of things was about 12 weeks and cost around $450 USD including all government fees and shipping. Total time on the Thailand side was about 3 weeks and cost around $160 USD for the translation and MoFA service.
Also, I did all of this from Thailand. The agency I used in the states did all of the legwork and sent the marriage certificate here via FedEx.
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