Can I change my visa to a multi-entry visa in Thailand using my US marriage certificate without registering at the Amphur?

Required​ to get marriage​ certificate​ in Thailand​ first ka.

IF you are married to a thai and got married in the states to USE the US marriage certificate you need to follow a couple steps

FIRST you need to get a blank affidavit filled out and notarized by ACS (american citizen services) at the US consulate (you use a blank affidavit because the consulate can't certify a marriage certificate from the US as those are issued by the states not the govt)

Once you have that notary paper you take it AND your US marriage certificate along with your passport to a translation service get it translated into thai get it certified by the Ministry of Foreign Affairs office here as accurate THEN you and your wife can go to any Amphur (city hall) in thailand and get a Kor Ror 22 form issued. <- that's the form you get when a thai is married abroad. It's THAT form you need at the immigration office when you apply for either a 90 day Non-O visa based on marriage OR a year long extension of stay based on marriage when you got married abroad.

there is no way to get a "multi-entry VISA" inside thailand based on marriage :(