We have never had a problem getting a wagon. But have never gotten one right off the shiip, but near the tram stop that you get off at, on left side of the path along with the wheelchairs.
In the last six months or so, there have been strollers too, along with the wagons, no worries about deposits or anything, just borrow for the day. They look like the older more worn ones that used to be on the ship, but perfect for a beach day and comfortable for little and big kids.
My dd6 fits in either just fine
We get strange looks when she is piled in one on way back to the ship, but she has tendon damage in one leg and at the end of the hot day she has too much pain and stiffness to walk the 7 minutes
I agree with Kcashner, keep your stroller sand free! I see lots of people dragging their strollers across the beach and wonder how much sand gets in all the nooks and crannies.
Sometimes in the a.m. the wagons and strollers may be a little damp so grab an extra towel.
Enjoy your cruise.