It will depend on the jurisdiction that you live in. Most states require at least one form of valid ID before a person will be allowed to obtain a marriage license.
Generally, a foreign national who entered the U.S. lawfully and with inspection may become married to a U.S. Citizen and succeed in a marriage-based application for adjustment of status to become a Lawful Permanent Resident (to get a "Green Card"). This is true even if the foreign national has overstayed a visa and is out of status. Of course, there are many more details that address eligibility. It would be wise for you and your fiance to consult with an immigration attorney who, after learning all of the relevant information about you, your immigration history, etc., would be able to advise about immigration eligibilities, options and strategies, and could offer legal representation in the often quite complex application process.
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