Not legally. Never seen it done before.
I have seen a lot of people set up and grill when spending the day on the beach. I have never seen beach patrol get onto people for it. However; I do not know if it is legal especially since it's turtle nesting season and they don't want anyone doing anything that will disturb that because of them so close to being endangered.
I can tell you the day we visited they were removing small cement block bbq pitts they were circular about 2 feet high about 5 feet wide? I believe the beach maintenance crew? Were doing that. Have a nice day.
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