I think Western Flyers were sold by Western Auto stores, not Schwinn. If I remember correctly, and I once made a monumental effort to buy a Western Flyer, they are like Huffys or Columbias-basic American made 10 spds w/one piece crank, welded frame, etc....
When I was ~13, I went to the Western Auto store where I grew up and ordered a Western Flyer 10 spd out of their catalog. The store clerk told me it would be there in one week. I came back on the day he told me to and the bike wasn't there. This went on for 5 weeks. Finally I asked for my deposit back, walked down the street to the BF Goodrich tire store, and bought a 5 spd w/upright bars off the showroom floor. I rode that bike alot.
You should be able to get parts at any bike shop. If the bike shop has been in business for a while, it's more likely they'll have the parts.