First, you have to realize that the spiders are there because of the presence of insects in your car. Spiders feed on many insect pests.
The reference to "Palmetto Bug" causes me to think that you live in the Deep South of the US--possibly in Florida--where they euphemistically refer to the native giant cockroaches as "Palmetto Bugs" ("Oh, we don't have roaches in the house darlin', its just some Palmetto Bugs" Yeah--right).
Anyway, the bugs would not be living in your car unless there is a food source for them. And, of course, the spiders would not be there unless there was an abundance of bugs. So--tracing the situation backwards, tells you that you need to thoroughly vacuum and scrub the car's interior in order to eliminate crumbs and other food-related detritus, and then vow to NEVER again eat in the car. Even the presence of empty soft drink cups, bottles, or cans could provide enough incentive for bugs to take up residence in your car.
After the interior has been thoroughly sanitized, I would suggest that you set off one of those time-delay bug bombs inside the car, with the windows closed. Then, after several hours, air out the car completely, wipe down everything to remove insecticide residue, and take comfort that your infestation should be ended.