Carrying tickets is a Dad-job in our family. He puts them all into a small ziplock (along with any fast-passes) and carries them in a fanny pack or his pocket. Only Dad is allowed to get fast-passes, and he's very careful not to leave passes in the machines. The kids complain about this and want to carry their own tickets, but I don't think they're old enough to be trusted with a $250 bill at this point in their lives. We'll have to adjust this system when the kids are old enough to branch out on their own, but for now it works.
Just for the record, keeping up with the money and the priority seatings is my job. Each child is responsible for her own spending money and her own water bottle (in a fanny pack).