There is NO free lunch, you will need many batteries and a much better alternator.
You could in fact attach a window mounted air conditioner like @Spehro Pefhany suggests.
TEC Thermo Electric Cooling
There are peltier elements that also generate a cooling effect compressor free. They are essentially a plate of ceramic postively charged, and one negatively charged. This creates a heat flow where one side is incredible hot and the conversely the other side is cold.
So you could mount these either inside your door, or perhaps behind the backseat.
Either way you would need to transfer the resulting heat outside of the cabin. If you don't mind an inferno in your trunk, behind the back seat. You could attach the cold side to the back seat and put giant set of heat sink fins in the trunk, with 12v fan to keep it cool.
If you were mechanically inclined you could also use water cooling. Install a second radiator under the hood and 12v pump and fan. Then the heat would get exhausted that way. If you don't have enough cooling you will break your peltier element.
here is an example:
Larger ones exist. However, if you purchased a bunch, say 20 you could have a setup where you control how cool it gets by turning some off. I have no idea how many you would need, but give these are 1.5 inches many would be needed.
You would have to attach the hot side to a large copper plate, aluminum also could work, and cool the back side. The down side to aluminum is it can't store any heat, and having used these to cool CPU on a computer I have come to the conclusion copper is better. On the cold side copper or aluminum would work, but make sure the 2 sides don't touch. The peltiers are less than 4mm in many cases so you will need something thin to keep them apart, depending on the total size of your device. Also a thermal paste will increase conductivity and heat/cold transference.
You will then need a large array of heat sink fins and several 12v fan to cool the back side. Either that or a water cooling loop.
The good news is all of this can be done with 12v-15v. The bad news is even 1 peltier element requires 7 amps at 12v. The fans and pump (if you go that way) will use very little compared to this. You can buy 12v 120mm computer fans cheap that use very little power. Even 10 of these is probably 90A with the pumps and fans. Your battery will die quickly.
In electricity, the smaller the gauge, the thicker the wire.
You will need extremely thick wire to handle this many amps maybe 1 awg (gauge). For simplicity you can connect up to 20A on 12 gauge wire so 3 of the listed peltiers. Then you will have to connect those wire bundles to even thicker wires and so on. Finally to a car battery, hopefully a 2nd car battery. In fact you will need many car batteries to keep this array powered for any length of time.