Many cars, such as the Vega, have no frame. They are called "unibody" cars.

The frame of the cars from the 20s & 30s were very narrow. your Tempest frame is probably too wide for an early car body to fit on. You will need to roll under some cars with a tape measure to see if they are even close to the size you need.

If you have a limited budget, cars from the thirties are tough to find, and harder to build. For your first project I would recommend buying a running car that you can lower, paint, upholster, put wheels on, and make it look the way you like. Even that will soak up more dollars than you think.

Once you have had some experience with modifying a car, you can try something a little more difficult next time.