Sounds like I'm not the only one with this problem! Thanks for all the suggestions.
I have surrounded the car with snap traps, poison bait boxes, live traps and an electronic trap. Peanut butter has been my most successful bait, but I think I'll try some apple slices just for variety. I also have a few mouse traps out. I've stopped chasing off the neighbor's cats.
I keep the hood up on the car, and the light on at night. I've destroyed any nests I find.
I think my downfall is feeding the wild birds. I keep the birdseed in a metal trash can, and now use a squirrel baffle on the pole the feeder's on, but some seed still ends up on the ground.
Here's what my carport looks like: