I have a 2005 Impala. The windows fog up on the inside when it is cold outside or it is raining. They really fog up bad, to the point you can't see.

windows fog up on the inside simply because of the hot breath of the occupants
this heat is cooled by the screen and condenses and so a fog is created
simple physics really - hot moisture + cold air =rain to over come this quickly run the air conditioner ( climate control) and that takes the moisture out of the air in the car and then the windscreen clears
you are going to say that it is too cold for the ac to run well at the same time run the heater until you have a pleasant temperature that is dry ( climate control--when it gets too hot the ac cuts in and when it gets to cold the ac switches off BUT the moisture that causes the fog on the windows is gone )

If it were a leaking heater core, the inside glass surface would have an oily film along with the fog.
It sounds like your fresh air intake is not working. You have to get fresh air from the outside to be able to get rid of the excess moisture inside the cabin. Or you have excessive moisture inside from a water leak that the system can't overcome.

no defrost?? does the blower work.. does the heater work at all..check fluids and add if low.. check fuses if blower dont work..

fogging up occurs when the water vapour ( Humidity) in the cab air hits the cold windows Remove the humidity by running the ac in conjunction with the heater . AC units to make the air feel cold by removing the humidity in the cab air allowing you to perspire more easily and you feel cooler. By running to two together ( climate control in some cars) the fog will quickly disappear.

IN COLD WEATHER USE BI LEVEL AND TURN ON HEAT ON AC AND HEATER MODE SWITCH AND HEAT MODE THAT WILL KEEP WINDOWS FROM FOGGING UP KEEP CAR WARM ON THE INSIDE WHEN COLD OUT SIDE ALWAY CARRY PAPER TOWER OR SOME THING TO WIPE WINDOWS WHEN FOG UP MOST TIME I HAVE PROBLEM WINDOWS FOGGING UP WHEN YOU HAVE COLD RAIN OUT SIDE.IF WINDOWS FOG REALLY BAD I HAVE TO TURN MODE SWITCH TO COLD MODE AND TURN ON DEFOGGER.

have a good look inside the vehicle as i suspect that their is a water leak into the car somewhere so its damp inside the car try lifting the mats and see if damp under the front carpets

to clear the interior of fog you must put the fan on recirc and #3 fan position.with the a/c on HOT.it will clear as soon as any heat from the engine is allowed to the heater matrix to dry out the interior.if you allow fresh air into the car with fogged up glass,they will get worse as the temp is warmer inside the car than out.(they collect condensation).

Yes. Unless you get fresh air you will fog up your windows. Your breath or rain on your clothes or snow on your boots will all cause the humidity to increase and this moisture will deposit on the non insulated glass. Kind,of like a glass of ice tea on a warm day you will get a condensation on the glass. The temperature difference is the cause. Try to run your A/C with the temp set to hot. This will act as a dehumidifier.
also check your pollen filter make sure it's not clogged.

It sounds like your heater core may have sprung a leak causing your moisture problems.

It may be set to recirculate the inside air, this is indicated by a symbol of a car with a swirling arrow inside, the button will be in with the climate controls. always use outside air if it is cold or raining and you will avoid fog