So we went with Styrofoam. It cuts easily with school scissors, and is relatively inexpensive. Plus I already had a ton of picnic plates to use up. Bonus.
Since my kids are pretty young, I didn't want to mess with expensive print making paint (which is really thick and soo awesome to work with), so we just used water based kids paint. Which isn't nearly as fun, but it did what we needed. And since it's not nearly as expensive as print making paint, I didn't have to freak out when they poured huge globs of it on their tray.
Some people recommend attaching the stamp to a block of wood or cardboard box so the kids have something to hold on to. We didn't. Ours turned out just fine.