The wagons are not allowed generally, but can be allowed for special needs. You will need to speak with a CM at Guest Services to get a wheelchair tag for your wagon. Make sure to always use it in the "push" position in front of you. It is not allowed to use a wagon pulled behind you. Just to be extra sure about this, you might want to give Disney a call before your trip to confirm that this will still be allowed.
They allow it because the wagons are being used as 'stroller wheelchairs' for a disabled child. It is no more headache for a CM to distribute a wheelchair tag for a wagon than for a regular stroller. There aren't that many in the parks on any given day that we've seen -- and I'm referring to DLR here. The 'SUV Stroller Wagons' were eliminated when Disney made the recent rules about stroller size. So the only wagons we've seen in the parks since then have had wheelchair tags clearly visible. These wagons were hardly clogging up the walkways or blocking traffic, and it was clear that the wagons were medically necessary, so there was no anger or confrontation that we ever saw. Maybe a raised eyebrow or two until the wheelchair tag became visible.
Businesses can limit or restrict use of specific mobility devices if it is considered a safety concern. That is why WDW does not allow Segways. I believe that since stroller-wagons are generally prohibited, one may need to have specific need for the wagon to use as a mobility device — as PP mentions, additional medical equipment that won’t fit on a standard stroller or wheelchair. I believe the safety concern of the stroller-wagon is the canopy blocks sightlines of both the user and others moving about the parks. There was considerable discussion about the Keenz in particular when the rules changed (was that just last fall? seem so long ago now!).
