1896 Ford Quadricycle, assuming that Marge works at the station and says "You there, fill it up with petroleum distillate, and re-vulcanize my tires, post haste!" Cut to next scene.
The current writers of the show could go a full season without being that funny, and in this classic episode, it's a disposable one-liner, like they had so many good jokes, they could just throw one like that in there just for kicks. Different times, friends.
Anyway, you'd think that vulcanizing rubber at the consumer level was some absurd thing that never happened. Well, that's what I assumed, anyway. Nope. Meet the Shaler 5 Minute Vulcanizer.
In 1925, your tires came from the factory all vulcanized and ready to roll, thankfully. But, should you need to patch the inner tube (yes, car tires still had inner tubes, like your bicycle probably does) you would have to vulcanize the patch yourself. I am not making this up. Here is a FaceTube video of a gloriously restored Shaler Vulcanizer. The guy seems pretty proud of it.
Yep. It's a clamp. Pretty much. It's job was to squeeze the patch onto the tire. The patch was permanently sealed to the tire with a little puck of fuel, which the motorist lit with a match. Presumably, the fuel puck also contained a bit of sulfur, which seems to be key to the vulcanizing process.
Good news, someone! The Shaler Co. is still around! Here's their page on their old vulcanizer product. What are they up to now? Sealing hockey rinks. Stick with what you know, am I right, people? The Shaler company used to be located at 2104 Fourth St, Waupun, Wisconsin. let's go see what's there now!