Hello Mary:
You should be able to do this by mounting a drawer in the base of the cabinet, and attaching a standard door front to it. However, the door may pull on you, and that might be bothersome and eventually cause it to warp. I showed something about my process with this here:
Shallow or 12" deep drawer for undersink pullout: will it work? Look for post #12 to show the comparison of short vs regular depth drawers.
I'm not sure if there is any other type of pullouts that would work in the shallow cabinets. Perhaps someone else will come along with a brilliant modification!