Looks like a thin aluminum plate on top of a freezer, so the first thing I'd put on a drawing board: maybe get a cheap chest freezer, cut a big hole in the top and replace it with an aluminum plate?
You would need a freezer coil pressed up against the metal plate. Maybe if you bought a mini-fridge with a little freezer in the top, you could take the freezer out and mount it under the plate. But if you're going to go through that much trouble, you might as well just get a compressor, some refrigerant and some copper tubes to make your own coils and condenser.
Maybe you could try getting some of those gel packs, sandwich them between metal plates and freeze the whole thing in the freezer?
Alternately, you could get some cakes of dry ice and lay a piece of metal on top of them?
To do it at home, just mix a few bags of ice with some driveway salt and lower a stew pot or something into it. At that point you're 2/3 the way to just making ice cream from scratch. But making it on a metal plate would be a lot easier than stirring it in the bucket, plus it looks fun so you could get kids to want to do it.
Homemade ice cream is worth the effort, but hand-stirred homemade ice cream is not.