Ah yes...the Schaller slip sheets. Those tuners are, I believe, made by Schaller for Fender. That friction plate design is used on Schaller's classic BLM series tuners. I deal with them all the time, because I modify Schaller BLM's to use on my Scroll Basses.
Note the thin steel plate that's under the black plastic. It's sticking up at an angle, and the longer side of it fits underneath the worm (the spiral gear on the key shaft). That plate is called the friction plate. It's made from thin spring steel. When you put the gear on the post and tighten it down, the gear pushes down on the friction plate, which then pushes up on the underside of the worm. That's how this design of tuner creates the friction drag on the key shaft; The outside of the worm rubs on the friction plate. The drag is controlled by the springiness of the plate.
The black plastic part is the slip sheet. It's made of thin Mylar or Acetate, cut to the same shape as the friction plate. It sits right on top of the friction plate, with its long end sticking down between the friction plate and the underside of the worm. It acts as a slippery washer under the back face of the gear, but its main purpose is to prevent the worm from wearing itself out from rubbing against the friction plate. That's why it's there. Over time, the slip sheet will crack or split, and the overhanging end will get jammed out from under the worm and tangled up in the teeth.
You can reassemble the tuner without the slip sheet, and it will function okay for a while. But eventually, the tuner will start grinding, and you'll see that the plating on the outside of the worm will be wearing and peeling off. The plating chips will then get into the teeth, and then the tuner will be grinding even worse and get hard to turn. Basically ruined.
I don't know if you can buy replacement slip sheet kits. You'd have to ask Fender or Schaller. You can make up replacements from some thin Mylar or Acetate sheet from a hobby shop. Cut them out with scissors and a knife. Or, stick a piece of clear packaging tape onto the friction plate and then trim around it with a sharp knife. I've done that few times. Good packaging tape, like 3M Scotch, is thin Mylar and is very strong and slippery.