What I would do is spray the vents with peppermint oil and then turn on the heat full blast.
My theory is,,, spiders don't like peppermint so the spider will avoid exiting through the vent thus remain trapped and the heat inside the vents will fry it.
BTW, I did spray my vents one time only with lavender oil to soothe my dog. Four years later, I can't smell it but my dog does take a quick sniff at the vents when I have the heat on. So I'd say the peppermint oil scent should remain a good while especially to repel the tiny critters.
Good luck!
Oh it’s a recluse. They are very common here. Old timers used to put crickets or lizards in their closets to eat them. The good thing is that unless you are cleaning out a closet you don’t have to worry too much about them because they hide. I keep those sticky flea traps with lights out all the time just in case. Having one in the car is a first.
My brother-in-law almost lost a hand from a bite from one. He was into junk/flea market sales and kept a garage full of junk to sell. He was sorting through some clothes and one bit him on the thumb. His thumb turned black and it took several surgeries and skin grafts but he kept his hand.