A car tire can be used as a propellant source with the proper adapters, however they 1) Don't have a lot of air volume, and 2) Don't have much pressure. The results is that you don't get a lot of painting done before you wind up having to reinflate the tire. I have a spare tire adapter, although I've never used it, and the instructions with it state that the tire must be mounted on the wheel. I'm not sure if it says that because of tubeless tires, or whether the inner tube actually has to be mounted in a tire and wheel. You might need a moisture trap, I'm not sure. It will probably depend on whether the place you use to inflate it has a trap on their compressor.
Other alternatives are carbon dioxide tanks. These are usually available for lease from places that refill soft drink machines or welding supply stores. I think some places lease the regulator sets, but some may require you to buy them. This is an excellent way to go, and is in some ways superior to a compressor. CO2 is perfectly dry and the source is perfectly silent.
You can also get a tank for inflating tires. These run about $30 at Sears or Lowes, but you'll need to add a regulator and probably a moisture trap as well. Additionally, you'll need some adapters and fittings to get the air to your airbrush.