Going by the information contained in Concealed Carry Resources section of USA Carry, Florida does indeed issue non-resident concealed carry permits/ license. I obtained the necessary paper work to apply for this permit but, when my state's permit law changed to allow a five year permit, I dropped all interest in the Florida license. Things may have changed by now but the possibility used to exist.
Florida Concealed Carry Permit Information
Florida Concealed Carry Quick Stats
Shall Issue
Permits issued to Residents and Non-residents
21 years of age required
Valid for 7 Years
Initial Cost is $112
90 Day Processing Time
Florida Reciprocity Maps
State of Florida Concealed Carry Homepage
Florida Law
Florida Firearms Forums