I believe the only baby item Disney routinely supplies for DVC studios are pack and plays-if one isn't in the room at check in they will bring you one. I've read that cribs can be requested at some Disney hotels as well but they don't have a lot (and are supposedly phasing them out) so if an actual crib is necessary people recommend those rental companies. I also recall reading that the villas with kitchens include a highchair so you may be able to request one for a studio. You will not find diaper genies or anything like that avail. I'd suggest packing plenty of the disposable bags they sell for your diaper bag and plan on running them to the trash room regularly.
We have been in many studios that high chairs as well as pack 'n plays so they are not limited to 1 & 2BRs. But if either are not in the room you can always request the item through housekeeping. You would have to rent a diaper genie (I never knew hotels carried something like that. Since DVC rooms do not have daily maid service you would need to take the dirty diapers to the nearest trash and recycle room.