Which one of these two is correct? Which one is more common? Or, if none of them is correct, how would you ask this question?
What's our wagon number?
implies the wagon actually has a number assigned to it.
What number is our wagon?
includes, but is not limited to, the first possibility. One other possible scenario is a parking facility with numbered lots. Imagine a family pulls into the lot and parks right next to a wagon that looks exactly the same as theirs. On returning to the facility, the son runs to both wagons but cannot tell them apart. One is parked in Lot 47, while the other is in Lot 48. The son excitedly asks his father, "Which number is our wagon, Dad?!"