Hi, and thanks so much for your question! The child care clubs at Walt Disney World only take children who have reached the age of four and are potty trained, so that isn't much help for babies and toddlers.
A service I have used in the past is called Kids Nite Out, and you can find them
HERE . This service takes children from infants up to age 12, and they will come to your hotel room or follow you into the theme parks. The service features sitters who are over the age of 18, and are CPR certified, with training in basic first aid.
The sitters we have used have all be very friendly, comfortably over the age of 18, and they have all brought some little things for our girls to do - coloring books or little toys. If you choose to have them watch your children at the hotel, you can give permission for them to take the kids to get something to eat. They do not take the children to the pool.
There are other area services, but I have only used this one. I find them to be professional and caring, and would definitely use them again.