Thanks for your question and welcome to the community.
Unless it's constructed extremely thick, corrugated cardboard is a very poor substrate to use for projects like this....due to it can fall apart quickly.
Worse still, is that the casters underneath would be heavier than the box itself. To give yourself a better surface for the contact paper, I would highly recommend you go with an equally thin but way sturdier substance that is wood-based.
You have a lot of choices in this matter, but I have personally found that luan or MDF plywood that is around 1/4" can work in situations like this.
Since you would build it with a sturdy surface, this means you can reuse and extend the life of the boxes far greater than anything in cardboard would do.
Depending on the carpet you have (how thin or thick the pile of it is) will determine if and how casters can work on it.
Only you can decide that facet of the project; check out the selection we carry at your local Home Depot of casters and see which sizes work right for you. For the size you gave us, you luckily don't have to use anything with wheels larger than 2" unless you want that.
There are also additional fasteners to hold the box in place, as well as a lazy Susan turn to allow the pivoting to take place.
Again, start with good materials and you'll have a sturdy box that can take the use of even the rowdiest of children.
Let us know if you have any further questions,