There are places in Key West that allow pets, mostly B&B's.
We take our's to CO when we go snow skiing...
We search for "pet friendly" places, like condo's and private homes....The site is VRBO.com, click on the area you are looking to visit, (by zip code even) and it will list the "paw" as pet friendly renters

Plus this is cheaper than using a travel agent...happy travels