Billy they actually made an in car cooler like you're talking about. The pictures in this ad might give you a few ideas.

Vintage Wright Mfg...Air Cool Car swamp Cooler working | eBay


I'm really not sure you're going to be able to get by without a reservoir and pump. For the cooler to work efficiently the pads have to stay saturated with water, and that much water will end up dripping out of the pads and need to be collected and recirculated.

It will be interesting to see what you come up with, good luck.