Chris with all of your experience turning bowls I don't see why you wouldn't be able to produce a fine stock for a 20 guage shotgun. It is not like inletting an action out of a much longer piece for an entire rifle action and barrel which is a time involved task.
But with minimal tools such as a good bandsaw, some rasps and files, a flat mortise chisel and various sandpaper you should be able to produce a very fine aesthetic and functional gun stock for your 20 gauge. Then when you pass the gun on to your son later in life, you can include the factual statement that you made the gun stock for the gun.
I really don't they it isn't something you couldn't do with your experience to successfully tackle. I have made Osage Orange Longbows and have learned when dealing with projects like this, that the best advice is, to work on it only when you are in the mood to work on it. To do otherwise is a great way to end up with a disaster....
Now some of the technical issues involved is that you would want to measure your son's length of pull to ensure the stock will fit him accordingly.
1. Measure this from the inside bend of his elbow to his first index finger.
2. Measure from the trigger to the rear of the action whereby the wood stock will eventually seat. Deduct this measurement from the total length of pull and you have the total length of the wood piece.
Next would be you could do some research to develop custom drop of comb and rise or drop of the stock's heel where it would seat on your son's facial makeup; to make a truly custom stock. Or you could simply follow the design of the original stock on the shotgun if he is shooting the gun great now with that original stock.
And if he is and you aren't familiar with the "custom" making issues of measurements and so on, simply use that stock as the template as you mentioned earlier.
Work on it when you are in the mood to do so and it'll look good.
Cannot wait to see you post a few pictures of it up here on the board in a few months or so.
Chris