Actually, I looked into a bit further. It turned out to be dirt simple. There's a hardcoded global variable which determines whether car-type vehicles have logistics grids. Any vehicle with logistics trash slots will have logistics enabled for it. Trash slots are 0 by default, and the car just doesn't have that definition (it's a nil value). Giving it 20 slots (same as the Tank and Spidertron) enables logistics for the car when the tech is researched. I actually threw together a simple mod for it: https://mods.factorio.com/mod/CarLogistics
But if you want to do it yourself, it's just a matter of setting trash_inventory_size for the car to any value above 0. All my mod does is this:
local car = data.raw["car"]["car"]
car.trash_inventory_size = 20
If you feel like implementing this option, please feel free. If you want to be fancy, you can even trawl data.raw for all car prototype objects and set trash slots if those haven't been set already :)