You can come on a visa exemption or tourist visa to look for a job, but you can’t start work until you have an employment-permitted visa/extension and a Work Permit.
Are you in Thailand or Singapore now? You need a notarized copy of your Singapore marriage certificate, a Thai translation. Then you need to get the Thai translation notarized. If you are in Singapore, the notarized copy can be done by a notary public and the last step can be done at the Thai embassy. If your wife is not confident to do the translation, you can go to Golden Mile or Beach Road area, many Thai translation service companies there.