It depends on the state. Most states require a photo I.D. If the person is from Mexico, s/he can go to the nearest Mexican consulate and get a matricula consular, which is a photo I.D.
i called a Mexican consular, and you have to have a photo id and birth certificate to get a marticula consolar.
*NO . but, if you can get a passport in your consulate or embassy, that's all you need. but if you better go back to your country and get married there... and start the process in the US embassy everything will be easier and fast.