I finally have a good design and process for building a cardboard pinball machine. I'll be working more to show how to make this later. One note of caution; the copper tape can cut you. I think this is because nothing flexes like when using copper tape with paper. When working with these cover the bends in your copper tape and just be careful where you grab things.
The shooter works really well. The flippers work well. Not shown there are two contact points with foil tape to trigger when you lose a ball. Overall really happy with the construction.
Parts breakdown; cardboard, copper and foil tape (with conductive backing), Popsicle sticks, pencil, rubber band, hot glue, LED's, 3V coin cell (like a CR2032), packing tape, and 3/4 inch metal ball. Not sure if there is a place to order fewer metal balls, but Super Magnet Man has this bag a 20 (they are not magnetic, just steel).