Short answer
May be possible, but not viable.
Long answer
Have a look at the Datasheet of a TEC1-12714S from Thermonamic Module. This is a 62x62x4.9mm³ PE, maximum current 14A, maximum voltage 17.2V. It is one of the most powerful devices on the marked.
The first problem a PE has is that it also conducts heat from the warm to the cold side, and this linearly depends on the temperature difference. While the heat pumping power is about constant, this means that the effective heat pumping power has it's maximum when there is no temperature difference. As soon as the difference rises, heat is also conducted backwards, and at some point, this "back-flow" equals the pumping power. So, the first diagram shows a linear dependency between temperature difference DT and effective pumping power Qc:
However, as long as it's really hot inside the car, the full pumping capacity will be available.
Note also that the maximum pumping power is about 140W for 14A and DT=0. If you have a look at the second diagramm, this current occurs at about 12V. So the PE consumes 12V*14A=168W of electrical power, already more than what it pumps. If there is a temperature difference, effective pumping power decreases while power consumption increases due to higher voltage. So, if t's really hot outside (40°C) and you want 20°C in your car, this PE will only have an effective pumping power of 100W.
Also, you have to dissipate pumping power plus consumed power at the outside, so 268W for the last example.
Up to now, we did not say much about how much power you need. A hair dryer typically has about/above 1000W, which heats your bath room only slowly. 100W may me much for a solder iron, but it's quite negligible for heating a room. The sun heats your car about 1000W/m².
I did not find numbers about car ACs, but is well-known that an AC takes some kilowatts from your engine. You may not notice it on most cars, but I do. I've only about 50kW, and on the highway with a slight slope, I sometimes call the AC switch my "inverse turbo boost". For sure, the AC takes about 2-5kW. As the pumping power can be higher than the consumed power (see this discussion, though not answered yet), you can estimate the pumping power to at least 3kW.
3kW, that's 30*100W, so 30 peltier elements each consuming 168W, being 5040W or 420A at 12V. Furthermore, you have to dissipate 30*268W=8040W of heat somehow.
I think, the numbers of my last paragraph show that it would be better to think of a compressor-based AC solution...
Best Answer
A peltier element is, in this order:
1.) a heater
2.) a thermal shortcut
3.) a thermal transfer unit
What happens is this: When you turn it on, the whole unit starts to heat up. But one side is somewhat cooler than the other. At first, one side feels warm, one side feels cold. Then the whole unit becomes hot because it is a heater primarily. The small difference of maybe 10 degrees between sides is not noticeable when one is 60C and the other is 70C. Source: [1]: http://quick-cool.de/peltierelemente/fragen-zu-peltierelementen.htm "FAQ Peltier German"