I like the rigid foam idea, but wonder of it would stand up to the outdoors.
My idea is to make a lean to shaped approximately like the front of the car, but about a foot bigger on all sides. I would frame it out of whatever is kicking around. Got a bunch of old PT 2x4s from an old deck which would work fine.
Anyhoo, frame it up, then put the tarp inside, suspended in place using light rope. Hang the tarp so that it creates an opening a little smaller than the car. Then lay as much fiberglass batt insulation as it takes to fill the space between the tarp and the frame. Then sheath the outside in painted plywood/rigid foam/what ever you have kicking about. Use galvanized wood screws so the sheathing could be easily removed. I would put insulation on the ground as well. I suspect rigid might work best as fiberglass batts would likely get wet. Cut out the insulation where the tires are.
The tricky part is making it air tight underneath. An inflatable plastic tube spread across the floor of the opening might do the trick. Or maybe just fiberglass batt insulation which would collapse as the tires ran over it, then expand behind them.
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