Chickapin Hill appears in the film's animated sequences, where Brer Fox and Brer Bear are seen concocting their plans to catch Brer Rabbit.
Chickapin Hill serves as the titular mountain, with the flume ride through the world and story of Brer Rabbit taking place around and inside the mountain. Brer Fox's home specifically comes in during the climactic lift-hill to the top of the final drop, where Brer Fox has tied up Brer Rabbit, who pleads to not be thrown into the Briar Patch, leading to the big plunge.
A backstory explaining how Chickapin Hill became flooded can be seen in a "Rabbit Tales" newspaper story in the queue:
Towering up and above everything hereabouts is Splash Mountain. Used to be that once upon a time, when most of us were still critlins, Splash Mountain was called Chick-A-Pin Hill. But that was then and back before a moonshining raccoon named Rackety, made a slight, but potent error. While mixing an experimental batch of brew, his juice producing still ended up being blown sky high. Many who were there at the time speculated that it was an overabundance of blueberries that caused the disaster.
This was unfortunate, not only for Rackety, but for the industrious Beaver Brothers, who had only recently finished construction on their new dam. Unknown to the Beaver Brothers, Rackety had built his juice still in the woods that backed up to their dam at the high end of the foothills. When the still exploded, the beaver dam burst forth, and all the water is was holding back, flooded the thousands of burrows, holes and tunnels that crisscrossed the inside of Chick-A-Pin Hill. From that time on, all the critters round here couldn't help but call this place Splash Mountain.