An airbrush only needs a source of air. As long as the air is within pressure limits, it doesn't care what that source of air is. You can run a hose from the tire on your car to an airbrush and it will work just fine (although it will probably smell).
Based on this, the answer to both your questions is yes. Walmart sells compressors; I don't know if they sell hobby-specific compressors (if they do they are probably Testors compressors), but a general purpose, garage-type compressor works just fine. An air tank can be rigged as an air source, and a lot of people do that. You'll need to add a regulator so you can regulate the pressure, but it's been done a lot.
Oh, and welcome to the forum!