Here in Far West Texas, the pronunciation is "shaps", for having to ride in the thorny chapparal - and lemme tell you, it's super thorny. If you said "chaps" like chapstick, people would look at you funny.
But really, you don't hear either word very often at all. Chaps in this area are almost always called "leggins." This is working ranch country. People of course know what you're talking about when you say chaps, but leggins is the only word used by the cowboys and ranchers.