The web site of the Florida's Division of Motor Vehicles has a lengthy table of documents required by foreigners wishing to get a drivers license in the state. Those who are APPLYING for asylum need to take their valid passport, I-94 showing entry date, and the receipt (I-797) issued by Immigration showing that the application was received.
Those granted asylum by an immigration judge need to take the original (and a copy) of the order. Asylees are considered immigrants and generally have an easier time renewing their license when it expires.
Family members who entered as refugees (including spouses or children of those granted asylum) need to show their valid passport and I-94 card with refugee stamp.