How do you fix a toilet after a cardboard crayon box has been flushed down?

If the box is big enough, it probably can't escape the built in trap of the toilet ,and will not get wet enough to disolve.

First, I would get or rent a toilet auger for a few bucks and gently try to catch an edge with the coil to pull it out. Then I would try to be more forceful ,then I would try to ram it to punch through it to get an area that holds so I could pull it out. If all that fails. I would drain the tank and bowl yank the toilet out lay it carefully on its side. If it is beyond reach of hand, auger or wire hangar run hot water in to fill as much of the inaccessible area as possible . If cardboard came out, I would fill the bowl to see if the the water was visibly coming out of the bottom to see if it drains normally.

Thats assuming it didn't wash down the closet bend of the drainage system. If no cardboard comes out and drains normally, than it probably is stuck in the first drain bend in the drain system. I would turn on the water supply and see if water drained or if the pipe fills up with standing water. If it fills or drains too slow enough to drain 1.2 or 1.6 gallons easily ,I would take the auger and plunge directly from the closet bend . Closet bends have a maximum of 6 feet allowed to get to their bend into drain, but it is almost always abut half that,you should have no problem reaching that with auger . Try to fish it out slowly, you don't want it to get stuck down the line. If you feel no resistance it may already be further in . I would run hot water in the system to try to dislodge it, getting it past bends. If it ends up in the mainline, normal usage should wet it down and erode it. If water still fills in the pipe, Dont use a small snake, as your liable to get it wrapped in a knot in the line. Use 1/2 " snake cable size or greater.That should knock out any cardboard from the sytem Rent one or call plumber if all else fails.