I agree with Joseph E. Meehan. A clogged evaporator drain tube is the most likely problem. The drain gets clogged with leaf debris and dirt, and the water that condenses on the AC evaporator has no where to drain, so it drips out the heater vents onto the floor.
Normally this water would drip onto the ground as you drive, but when the drain tube is clogged it can't escape. The drain tube also serves to drain the HVAC system during rain, so if you don't clear the drain you might get water in the car next time it rains.
After you have cleared the drain you will have to remove the padding from under the carpet and dry it outside the car. The padding is like a sponge, and will hold water even if the carpet feels dry. This trapped water will allow mold and mildew to develop.
You may have to raise the car to reach the drain tube. DO NOT crawl under a car supported only by a jack. Use ramp or jack stands.