a few random thoughts…
Using, as we did last year, a chassis base of 3/4″ baltic birch plywood, gives you a good foundation onto which you can attach the mechanicals. For axle to base attachment, we used one of the methods shown in soapbox derby construction manual. The plywood base itself is not a stiff enough structure for a good racer, but with the sides attached the structure is very stiff (in essence a monocoque).
Remember to draw an accurate centre-line on the base, very important for accurate alignment of axles.
The picture below is of the car mostly assembled but not painted. You can see the foam nose (glassed over) and the extra bit of side material scabbed in. The plywood bottom was 96″ long, because the sides curve, an extra bit of 1/8″ ply is needed. The foam nose brings the overall length to 100″ (which was the max last year).
The blue top is simply 2″ foam, carved to shape and then glassed over. Obvious to point out here that polyester resin would melt the polystyrene foam, you have to use epoxy.