I have a van that I don't trust to take on long trips, but Geico doesn't cover rentals unless you buy full coverage. Unfortunately, they won't let me buy full coverage because my vehicle is not worth the risk.

You can purchase the physical damage coverage from the rental car company. The good thing about doing that is that there is usually no deductible so, should you have an accident with your rented vehicle, you don't have to worry about any additional out of pocket expenses. You should however, contact your insurance company and ask them if they have a coverage for the rental car that covers the expense for the time it is out of service after an accident.