While I was researching this for an hour or so, Ray provided the probable answer. A search of my library found nothing on the .22 Wingo Shotgun, evidently some sort of single-shot lever action firearm. No Winchester catalog I have from the 1970s or 1960s mentions either the gun or the ammunition. It is mentioned in the current edition of COTW, on page 504, but they don't even show a picture of the round.
I was active in the Gun Industry in 1970, and while the name "Wingo" was known to me, I don't recall ever seeing anything about the guns or ammunition. They certainly were not offered to us, and we were the biggest Winchester retail dealer in Northern California, I believe,
"American .22 Rimfire Boxes," (The Sam Stillwell Collection), by Robert Buttweiler, shows a picture of the box on page 190. He does not date it. Bary Gracia's book, "The .22" Rimfire Cartridge" shows the same box on page 17, and says that the system was introduced by Winchester in 1970. The Winchester commercial catalogs of 1969, 1970 and 1971 make no mention of gun or ammunition for "Wingo."
Ray mentions it was more of a shooting facility, and this squares with the lack of information in the catalogs. It is probable that this system was never offered to the individual buyer, and therefore appears in no commercial catalogs. There was probably a brochure fore the gun, ammo and the range setup for running this enterprise offered by Winchester to prospecitve clients for the Wingo System, but I have never seen one. Gracia says "The firearms were available only at approved Wingo shooting centers. Made only by Winchester-Western and shortly discontinued, the brass case cartridges are loaded with 2.1 grains of ball powder and approximately 112 #12 shot to a M.V. of 950 f.p.s."
Thiese entries and the lack of information contained in commercial Winchester catalogs of the era it is supposed to have been offered, put Ray Mekata and Barry Gracia's book squarely in agreement with each other, and based on both men's knowledge, I would say it tells the story!
If anyone has a brochure for this system, please consider a short article for the IAA Journal centered on the brochure. Now that the question has been asked, I think all of us would like to know more about this subject than we seem to.