Thank you to the people who sent me feedback about this project. It is much appreciated.

After spending some time pondering and researching I have a design that is a bit more refined than the diagram in the description, not that that's saying much.

1. Bag input mechanism - The bag is fed into a tight set of rollers. The rollers are weakly driven and if something too thick is inserted, such as a folded up bag or a non-bag object, the machine stops and spits it back out. Ideally the bag is fed in flat and straight. After the rollers the bag is wrapped tightly onto a thin metal pin. Exactly how to make that happen is still a challenge I'm thinking about.
2. Heating tube - Once the bag is wrapped tightly on a pin, it is inserted into a steel tube and the pin is removed. A steel piston presses the bag down into the end of the tube, which is still cool. That end of the tube is wrapped in the coils of a relatively low power induction heating circuit which is then turned on. The tube is kept at about 180C until the bag is melted. I'll need to experiment to see how long that should take
3. Injection molding - When the plastic is molten, the piston pushes it through a nozzle into a disc shaped mold. One side of the mold has a small opening that excess plastic escapes through. Once the plastic has had a chance to solidify, the excess is chopped off and the mold opens to eject the disc.
4. Entertaining apparatus - This is still a topic a lot of possibilities. There have been some good suggestions, such as esot.eric's plinko idea where there is a chance the disc could be won as a reward. Another idea is a Rube Goldberg type contraption that is just entertaining to watch. Another is an interactive game where the person tries to make the disc fall into a target or... well, I'm still thinking about it. I want to have a good brainstorming session/log about this, but haven't found the time.

One concern that was mentioned by Remus is that heating the tube will cause trouble with the fit of the piston due to thermal expansion, but a quick calculation shows that the tube will only expand roughly 0.002% over the relevant temp. range which is small enough that I'll ignore it.

So how about those first steps I mentioned. Well, I've been researching small induction heating designs and gathering parts to make a prototype heated tube. Then I can get an idea about the feasibility of melting a bag in a tube. Sorry I don't have a working schematic or pictures yet, but I'll post them once I do.