Vortex tubes aren't very efficient at all. They only use them when compressed air is handy (like supplied breathing air, or a shop) to get more cooling than the air alone. They're light and simple, but loud and take a lot of air.
A commercial source of them lists BTU/h cooling
here. The biggest they have is 6000 BTU/h... the A/C in my Civic is listed at 14800. It also requires 100 scfm, which would take a huge (~30hp) compressor. So standard
A/C is about 10-20 times more efficient, not counting the air-drying effect.
Thermo-electric coolers are also less efficient than refrigerant systems. Also, ice has more cooling for the same weight as a battery, so if you want a rechargeable system, just use a cooler full of ice.
I think the fan seat cover is the closest you'll get to all your requirements.