Are any tail wheels on small planes made of hard, rounded, plastic-like material?
You can see that sort of thing on ultralights that land at 25-30 mph where they may use a solid once-piece wheel made of some elastomeric plastic or rubber. Smaller airplanes however, will use a rubber tire on an aluminum wheel with proper bearings, due to the very high rolling RPM once you are landing above 40 mph (or at least a bushing that can take high speeds and loads).
The smallest ones usually have solid rubber tires and once you get to a certain size, say on airplanes over 1500 lbs or so, they will be pneumatic tires. For example, an Aeronca Champ usually has a solid rubber tailwheel, but it's heavier and faster Citabria descendant has a pneumatic tailwheel tire. You can browse typical tailwheel tires here.