I don't know if Home Depot or Lowes has this, but at Menards, in a seperate inside section from the outdoor lumber yard they have a "good wood" sectoin where you can buy small 2x2, 2x4, or 4x4 pieces of plywood in different thicknesses with different types of veneer that is presanded on one side, so you would only have to cut it and sain it. I know that at the Mernards I go to you can get oak, maple, and sanded pine, something like that you might be able to sain to match the set, or you could have a two-tone cabinet for that 50s look.
This "good" plywood will cost more than regular pine builders plywood, but will look a lot better in the end.
Applying Veneer takes some practice, and you would probably have to go to a specialty wood store to find something to match your set.
Good luck with the set!
..Is there any chance you could get a cheap 35mm camera to document the cabinet? It would be really nice to get before and after shots.