The 1" dowel is for the dispenser. To figure out the length push the dowel all the way into the nose (arm) hole until it bottoms. If it is really tight you can set the kids up on the lathe and a piece of emery cloth and get them to turn it down until it spins smoothly in the hole. Keep an eye on them that they don't try to use the disk sander or belt sander. Disastrous results with dowels spinning at outrageous speeds.
Once the dowel is in place and spins easily you can mark a cutting spot depending on how much they want sticking out. Take this moment to also mark the center of the dowel through the top hole. This is where they will drill the 3/4" hole for the gumball to drop into.
Cut the dowel with a handsaw. Don't use the bandsaw or it will invariably spin a piece of dowel. not good.
Mount the dowel in a drill press mounted with the 3/4" forstner. Drill in about 5/8" of an inch. You want to go deeply enough that the gumball doesn't stick out and stop the dowel from spinning and spitting out the gumball.
The kids love it if you explain what happens if you drill all the way through... you gently turn the dowel and OHNO the gumballs rain out all over the floor. Feel free to exaggerate at this point.
It was pointed out to me that if the dowel is pulled on it would rain gumballs! Great point! The fix as suggested is to drill a small clearance hole in the back of the block so that a screw can be pushed through and screwed into the dowel... make sure it isn't too tight or the dowel won't turn. Thanks for the suggestion jonny!