This question about how yodeling, which seems to be an Alpine Swiss tradition, got connected with Country and Western Music, particularly the singing cowboys, has puzzle me for years, as well. Of course, other traditions use the voice break as well- Hawaiian singing uses it, and, there were a lot of Hawaiian singers touring on the variety circuits during the twenties, (and many recordings, as well) and the commercial singing cowboys (and brakemen) may have borrowed from them, and there were probably Alpine Yodelers touring as well--the Alpine Yodeling is fairly elaborate, but the cowboy stuff tends to be fairly simple, as does the Hawaiian--Of course, African/American and African/Carribean music has a tradition of using voice breaks, and this would be an obvious source, too.As far as I know, it wasn't an element in real cowboy singing--they might have sung phrases like "Ti-yi-yippie-yippie-yay" but there were no voice breaks involved--Of course, popular Cowboy singers often were just pop singers with a horse--
I used to do a bit of yodeling when I was younger (it only works about one time out of ten nowdays), began by copying Jimmie Rodgers "He's In the Jailhouse Now" had a particularly interesting(but difficult) yodel to it. I learned a version of "There is a Tavern in the Town" that had a lot of yodeling in it, I also learned a version of the Hawaiian Wedding Song--but it has been a while since I could manage either of them--