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The cribs offered by Disney are Graco pack 'n plays. On our last 3 visits to WDW, we've found one in the room, folded up in the closet/sink area, even though nobody in our party requested or needed it. I don't know if they were left over from prior guests, or if they're kept in the rooms as part of the standard amenities now. I would add it to your reservation, as @sxeensweet suggested, and confirm at check-in that you need one, so they can deliver it right away if they haven't already.
And forgot to add DVC rooms if staying in one, already have them in the closet. Have seen one every trip in the closet when we go. Just got back this week and Kidani had one as usual in the closet folded up.
We pack our own pack and play sheet, WDW does supply one but we like our own. Make sure you mention it at a heck on, we have had to wait a bit to get one. Not a big deal for an early arrival but if you are arriving at night and your baby need to get to sleep it could be an issue.