Double check the Florida Wildlife Commission web page for current laws, rules, and regs. Most WMA's are open seasonally with NO HUNTING in off seasons, and seasons being Fall Deer and Turkey and Spring Turkey, all seasons now closed.
Private land hunting is 24/7/365 for feral hogs, and perhaps Nutria, although I have never hunted the latter, but double check the FWC game regs. Nutria, being waterborne animals, and center-fire and rim-fire rifles onto water bodies bodes ill for ricochets, perhaps the shotgun is more appropriate, although ask your land owner.
I have no insight into who to approach for permission. Others will have to answer that.