The Koza web site shows several methods for making a worm for an existing plastic gear. The beauty of the worm drive is that it can produce a great reduction in the rotational speed of your system or allow a much higher torque to be transmitted. Here we are discussing the worm component -- the screw-like element of the drive.
The first method (top) involves winding what looks like copper wire around a screw to form a helix. The trick is probably finding the right gauge of wire, then the right size screw thread. Still, if you've got a few gauges of wire and a bin full of miscellaneous screws, it doesn't seem like it would be too hard to do. I do wonder about the copper wire keeping its shape over time.
More robust solutions are offered in picture form (bottom). For the second solution, a screw is used as the worm, engaging directly with the
teeth of the gear. Here I worry a bit about the screw threads
actually cutting the plastic gear over time.
The third and final method is my favorite. A steel spring is used for the worm to drive the gear wheel. Simply take your plastic spur gear to your local hardware store or spring supplier and try a few out until you find one that meshes well. Though the image shows the spring alone driving the worm gear, you might be able to have a dowel or other axle material running through the center of the spring. This would make the spring easier to attach and provide more rigidity. You would just want to be sure the teeth still engage well with the screw and that the axle doesn't interfere.
Automata maker Dave Johnson makes good use of a plastic drywall anchor to serve as the worm on his piece entitled Little Man. You can can catch a glimpse of the drywall anchor driving a wooden worm wheel in the video of the piece.
Thanks again to the Koza web site for the first three techniques. Check out their site for several other interesting mechanical solutions.