Mike although I agree that streaks in glass may be a problem. It's a relatively small problem, and an intermittent cosmetic one at at. In he long run it's probably cheapest to just buy a lot of glass and sort it carefully.
Anyway regarding the PVC:
PVC as it's typically produced is not going to be clear. It's mainly amorphic with very limited %chrystallinity, basically the PVC chains are randomly arranged in the material like spaghetti, not arranged in a pattern like ramen noodle blocks.
Amorphous polymers are generally opaque, and crystalline ones generally clear.
Please understand this is a massive generalization and many materials are somewhere in between the two.
But on the other hand, lots of people have made PVC bows.
If I had to choose a polymer to make a backing material it would almost certainly be UHMWPE. The same stuff high performance string material is made from. It's used for tons of stuff, we'll understood processing technologies exist, and it's ridiculously strong. Is it clear? Maybe if you have fibers of it imbedded in more plastic like the way glass lams are made.
Cheers.