That was meant to be "water hose".
I actually used water hose to fix my flat on my bicycle when I was 8.
Oh, I misunderstood the weight issue on the score, got it backwards.
That means you can cut wheels from any flat material that doesn't flex too much. That way they can cut into the sand a little to get a truer track line, then glue a ridge around each side of the wheel to limit depth, increase traction, and limit friction. Don't know how deep the sand is going to be. Replace the the gravity drive with a rubber band motor. You still want a measured string wrapped around the axle that put a well defined travel limit on the cart and operates the firing pin. The gun would be close to the same except using a straw and stick instead of pvc with regular rubber bands. The straw can be glued to a stick for strength and stiffness while giving it a base to mount it with. Some kind of gun barrel is needed for accuracy. Like a pea shooter.