Here's a video of a 1947 plastic (vinyl?) recording stall.
That's very cool! Near the end, you can see the label on the disc says "Caution Inflammable", so chances are high it's made of cellulose nitrate. That's a good material for cutting grooves in: not the most stable, but it should last for 50 years if not mistreated, and wouldn't wear out too fast. I wonder if anyone still makes blanks like that, or when they're gone they're gone?
Polycarbonate can be lathe cut, you don't need a laser. Laser cutting would be hard to control.