Hi Bethany.
I have built a few different versions of a digital booth -- there are many ways it can be done. Software is the trick, as most digital camera (webcam) / computer / printer combos should work fine. Of course you have to think about the variance in image quality depending on what kind of camera you use, but typically you are printing so small, even a webcam should be good enough. Some cameras come with software that lets you program applications around a library of functions the camera manufacturer gives you. Logitech used to have a set of tools (SDK) that they made freely available, as did Canon. I use the Canon SDK for my camera control then a custom-designed C++ app that drives the camera, image manipulation, and printer control. There are probably easier ways to do it -- depending on your interface constraints.
-Tim