Many do at the outside tables. I think you can bring one to Island Grill, not so sure about Morada Bay.

There was a couple at Mile Marker 88 restaurant that had their labradoodle with them.

Marker 88 is not consistent with their dog policy. Friends of ours were told dogs were not allowed. Lor-e-lei, before Hurricane Wilma, had permitted dogs. After they rebuilt, they banned dogs. Now, someone told me they are allowing dogs again (NO DOGS sign remains) Island Grill will happily seat you and your dog on their beach.

While you probably don't plan to stop in Key Largo, Big Chill, Breezers, Snooks, Sharkeys, Buzzard's Roost (outside) have all allowed my pitbull.