We went last May with our 8 month old. It was a great trip. He was actually easier at 8 months than he was when we took him again at 14 months. Definitely use the baby care centers. They have nicer areas to change and feed and are a great place to rest. Try to stick with the schedule as much as possible. One day we just carried him around for the nap instead of going back to the room, and we regretted it. Every other day we stuck with the schedule of naps and he was able to stay out later for EMH and sleep in a little later than normal. We bought a fold up stroller at Target for $30 and we could not have been happier. The umbrella strollers are not enough, but you don't want to bring your nice one from home or rent one of Disneys. The Disney ones are too big for a smaller baby and are plastic and get hot and sticky. The stroller is very helpful in the airport, too. Make sure to bring hats and sunscreen. Also, use the rider switch to enjoy rides you can't ride with the baby.
They have pack and plays you can request in the rooms, but I am a germ-a-phobe, so we requested the bed side rails and packed all the pillows around the bed. He was fine since he wasn't too old yet. Every other trip we have brought our own pack and play.
In regards to supplies, we wanted to pack light, so we rented a car and went to Wal-mart and bought the diapers, formula, water, etc. that we needed. We did not drive to the parks, but it was helpful to have the car to go to DTD and dinner reservations at other resorts. If you don't want to rent a car you can purchase those supplies at the Boardwalk store and Wilderness Lodge, but they are much more exspensive and not as large quantity. They have back ups you can use in the baby care centers, too. You can also call in an order at the Garden Grocier, but they have limited brands.
My final thought is to make sure to pack for rain. When we went last May it was monsoon rain every day night and day. Luckily I had packed several long sleeves and pants and his rain jacket with a hood and a hat. If we had not had that it would have been much more difficult.
Have a great time! It is so fun to take your little one to a place you enjoy. They won't remember it, but you will!