You can go to the Clerk's Office (link below for more info)
From a websearch in regards to Marriage in Florida;
All U.S. Citizens and Residents must provide their Social Security Number.
Non- U.S. Residents may provide an Alien Registration Card, Driver License, Passport or any other legal form of identification if they do not have a Social Security Number issued to them. Requirements for getting married in Broward county include a second form of ID if a U.S. Social Security number has been issued. The following forms of proof are accepted: original Social Security card, US Military Card, original printed paycheck stub, W-2 form with applicant’s name and social security number, SSA Medicare card, Third Party Notice or Future Benefits Earning Statement from SSA with the complete social security number.
County Clerk Directory;
http://dlis.dos.state.fl.us/fgils/coclerks.html
A little trivia: Tennessee only recently begin allowing people to obtain a licence without a SSN. They had 'imposed' the rule presumably to track deadbeat parents, but it was seen as 'profiling' moreso geared to the Hispanic community. Interesting enough, it was a lawyer who applied (her Fiance had no SSN and was refused) for a marriage licence who really started the ball rolling to change the directive. Waiting periods, costs and such vary from state to state, but the information is readily available online.