What would make my 2002 kia rio stall and shake ...

There could be a failure in the idle speed advance system.  Usually the air conditioner compressor is activated when the heat is turned on.  The idle speed needs to increased slightly to compensate for the extra load.  If you have a 4 cylinder engine, is is more sensitive to extra load at idle.  There could be  a vacuum leak in the heater control system but, this usually causes the engine to idle faster and the computer will adjust it somewhat.  Try turning the air conditioner on and see if the same problem occurs

OldDad gave a good answer because the IACV  (Intake Air Control Valve) is affected by vacuum and it adjusts the idle speed of the car by adjusting the amount of air to mix with the fuel and sending information to the ECM (Engine Control Module). There are sensors that attach to the IACV that adjust the speed control  when the air conditioning puts a load on the engine when the AC compressor requires . Most of these same controls are activated during the heat cycle. If there is a faulty valve or hose in the HVAC system then it could cause a vacuum leak and that would cause the car to run rough because it could not only screw-up the air fuel mixture it could more importantly drop the engine vacuum. It is the engine vacuum that draws the gas/air mixture into the combustion chamber so that the cylinder can fire . Bad vacuum equals bad performance. By tracing the vacuum lines related to the HVAC system you will most likely find a cracked or disconnected line or even a defective valve. You can check the engine vacuum to see if it drops when the HVAC is engaged by placing a tee in a vacuum line off of the intake manifold. A healthy engine should have between 18 and 22 inches of mercury. Just look for the needle to be between 18 and 22. If it drops below that when the heat or a/c is cut on then you know it is a vacuum problem specifically in that system. Fuel pump/vacuum gauges are pretty cheap and can be found at Advanced, AutoZone or O'Reily's.

Good luck,

Kaiser Willy