I don't know 100%, but at least a diaper genie shouldn't be a problem. My dad has to request one for the Depends he needs since his prostate cancer surgery (he's totally clear now) and does it when they meet their stateroom host. (He does it day-of mostly to save any confusion since there are no children in their cabin,)
Pack and Play you could probably request day of and be fine since while I doubt they have one for every cabin, the odds of every cabin needing/requesting one are probably about diddly over squat.
I'd say if you're really worried about it, request it in advance - but I doubt day-of would be a problem.