Quick answer, buy a small ice maker unit designed for a wet bar or small outdoor kitchen. The units I worked on was basically what you are trying to build. By the time you build your own unit you aren't too far from the price of a unit already made.
What follows is the long story of why you should chalk it up as a good idea but too much crap to deal with.
I've been out of the small appliance repair biz for about 10 years so the information may be dated. I would double check location and routing of the evaporator lines (cooling coils), in most of the small freezer chest units I worked on the evap lines ran on 4 sides embedded into the walls.
There was no real way to really tell just where the line really was inside the wall. Sometimes, there is a sort of layout of the refrigerate lines posted in the handbook or on a sticker pasted the the backside. Many times, there was not. One wrong hole drilled into the cabinate at the wrong place and you are screwed!
Due the embedded design of the evap coils within the walls I doubt that it has a defrost unit installed. For a self-defrost unit it would need to have a heater and timer to induce melting of frost build on the evap coil. Typically a defrost heater is only used with a freezer/refrigerator if there is fan motor used in freezer compartment.
A small chest type freezer should handle freezing of the ice within the ice maker with little problems, perhaps even two ice makers.
A couple of things to bear in mind. There were two types of ice makers on the market when I was in the bus. A flex tray type and a heated mold type. The flex tray used no mold heaters but was hell on trays which needed replacing sometimes as few as every 6-8 months or as long as a couple of years depending on useage and water supply I suppose.
The second was a heated mold that allowed a small heater to turn on at the end of a freeze cycle to loosen the ice in a mold for the rake to remove.
Either version needed a minimum of three wires for proper operation. First was a hot power lead,second a neutral line and third, a control hot power lead to supply the water inlet valve with power when calling to fill the mold. The water inlet valve of course would need to be mounted to the outside of the unit or two units if you are installing two units.
One 1/4 inch supply line is all that is needed for one or two ice makers, the unit typically only uses 6 to 10 oz. of water to fill the tray. As long as you have good water pressure coming through the main 1/4 inch line you will be alright. By the way, most home appliance water-inlet valves need at least 20 pounds of pressure to operate correctly. If lower the diagram will not seat properly and will allow water to seep past it causing a leak.
I must confess to scratching my head about the low cost of the ice makers you seem to be getting as back 10 years ago the wholesale cost of one was about 70 bucks or so for a complete kit. Be that as it may, I would encourage you to get a firm price of a unit before committing too far along. Be a shame to buy a small freezer only to find the ice makers costing you 125.00 a piece.
The ice cubes will more than likely will eventually turn into a massive chunk of ice as time goes along. This due the opening of closing of the door ever time someone needs to get some ice out. I could be wrong about this but its something I would consider.
Ever defrosted a freezer with 50 or so pounds of ice froze together? Quick way, a water hose and a lawn needing a very deep watering. Slow way, a few days and a bucket to dip the water out once it has melted.
Even if you aren't too concerned about the ice buildup or the electrical connections that will need to be made to each ice maker I would forgo the idea due the evap coil placement and the all too possibility of hitting a internal refrigerate line in the wall.
Now if you want to talk about adding a beer keg and tap to an old refrigerator, I might have better news for you.