I have no first hand experience with rentals in the KSC area, but a google search of "Cocoa Beach, condo, rentals" came up with lots of hits.
I wanted to put in a plug for a trip to Playalinda Beach at the Canaveral National Seashore. It is one of the most beautiful, unspoiled beaches in the country. We grew up in Titusville at the time of the Apollo missions and had NO IDEA the treasure we had. All we knew was that the drive home through the incredible marshy ecosystem with lots of water fowl took forever when we were sunburned and sandy. The beach terrain changed constantly, the waves varied from placid to frightening, the shells changed all the time, no rocks!, great sand, etc.
There are no facilities but it is worth a trip and a nice contrast to Cocoa Beach (or New Smyrna, or Daytona, etc.) I think it is closed at the moment because of the terrorist attack, but hopefully will be reopened by November.
Hope someone else can help with the rental.