Ahoy, Christopher!
Thanks for stopping by with your question.
While there is not a dedicated Baby Care Center on Castaway Cay, you'll find plenty of shady spots available around the island. Lounge chairs and large beach umbrellas are provided free-of-charge on the Family Beach, and are plentiful. If you want to choose a spot that's most private and quiet, I encourage you to head off the ship early for your choice of locations. It's also a short tram ride/walk back to the ship if you'd rather be indoors for a bit while you feed your little one.
There are beach-ready strollers and wagons available on a first-come, first-serve basis when you reach the island. If you're in need of transportation, make sure to grab one on your way to the beach. When needed, there are changing tables provided in the restroom facilities around the island.
Castaway Cay is so beautiful, clean and serene. Our youngest had just turned two on our first visit to Disney's private island paradise, and I have nothing but wonderful memories of our day there! Make sure to bring a camera to capture special moments - it's the perfect backdrop for beautiful beach pictures!
I hope this helps with your planning. Please come back and visit if you have more questions before you set sail! - Diane W.