So some further arm chair engineering. I took your fridge cubic capacity and generated an estimate of the dimensions from the photos. I have used 14" wide, 22" long and 19" deep.
Soft water quality 3/8 copper tube is available in coils from your standard big box hardware
stores. Link from Lowes below as an example. Recommended manufacturers bend radius is 11/2 " , link to tube bender example below. Use one to ensure you do not crimp the tube when bending. Making vertical S style coils on the sides does not give sufficient tube volume capacity of chilled water.
However CaptTom's suggestion of running coils around the inner wall surface the same as a beer
chiller wort is the way forward. With the above estimated internal wall size, and with the 11/2 " radius bends at each corner, one complete 3/8" tube circuit would yield 7.5 fluid oz (222ml), less than an "standard" 8 fluid oz (250ml) glass of water.
You would need to stay away from between the wall and mounted condenser, as that could freeze the water, but if you ran 6 coils below the condenser and then out to your faucet, you would obtain up to approximately 5 glasses of cold water in one pour.
With the dimensions I have used, you would need around 40' of tube, including the external connections. Insulate the copper or plastic on the way to the faucet. Have the water enter at the bottom of the fridge and out on the corner where you are placing your faucet.
Your needed length of copper tube (and hence volume capacity) will depend on your actual fridge dimensions, number of coils you can install, connection point to potable water system and faucet.
I would install a water isolator on the copper tube before it enters the fridge, so if you damage the tube in the fridge, it can easily be isolated. You will find compression
connectors for the tube at hardware
suppliers. If you are typical boat maintenance
handy, then it's an easy one day project
once all the materials and tools are assembled. Links here to examples of tube and bending tool.
Happy cool water drinking.