Lets see:
The cabin air system will not allow the rates of flow you need to feed a turbocharged engine, or even a normally aspirated one. The cooling element is a finned matrix, like a radiator, but quite small and all contained behind the dash. The amount of ducting and pipework you'd have to do would be enormous.
The air cooling system works by refrigeration, pumping heat from one place (the cabin air inlet) to another (the condenser mounted in parallel with the intercooler and engine coolant radiator). The clever thing about refrigeration is that it allows you to cool things down below the temperature of the outside air. It expends significant amounts of energy doing this, which results in more heat - so you need to dispose of more heat from the refrigeration system than just what you suck out of the cabin air. All that heat is disposed of through the refrigerant condenser into the air next to your intercooler (if you have an FMIC).
Given that we are trying to cool down air that's going to be heated up by the turbo, an intercooler is a much better option. That's where the inlet air is hottest, and the bigger the temperature difference the easier it is to transfer energy.
Even for a normally aspirated engine, you would exceed the cooling capacity of the refrigeration system very quickly, so even if you ditched the cabin air system and rerouted the refrigerant to a much larger, well-designed for free flow cooling element in the engine inlet, the gale blowing past it would mean that you'd only get a small temperature drop.
Intercoolers, or charge air cooling by any other means (i.e. water/methanol injection) are a much better idea, more efficient and without the large weight penalty of the refrigeration system.
If what you were thinking of was to route the cabin air system into the airbox as an additional source of air at a lower temperature, see above - the amount of air you can provide is far less than the engine needs, and the overall reduction in inlet air temperature would be minimal, and certainly not worth the effort.