2001 Accord EX and the front passenger carpet is soaking wet

The two most common causes are rain water leaking in or the ac drain being clogged up. I would vote for rain water if the entire carpet area is wet.
Could be coming in around the windshield, or one of the door gaskets.

DRAIN HOLES IN BOTTOM OF DOORS? UNDER BONNET BELOW WINDSCREEN CAN COLLECT A LOT OF STUFF

Blocked A/C drain or leaking heater core. Are you going through coolant?

check your window seals and seals by defrost vent. One is leaking

sounds like you have a bad heater core

If you've been using your air conditioner during this time, and it hasn't rained at all, then the small drain at the bottom of the a/c evaporator core may be plugged. When you use the a/c to cool the car the evaporator coil [located next to the heater core in the ductwork under the dash] is very cold and moisture from the hot humid air condenses on it as a result. This condensation dribbles down the core and accumulates in the plastic housing under it where it should then drain out a small hole and drip onto the ground. If the small drain gets plugged, which happens quite often, then the water fills up the housing until it overflows and then runs out and soaks the carpet on the front passenger side. The drain is located on the engine side of the firewall, passenger side, down low where the firewall begins to curve under. Usually a puff of air will clear the clog and the drain will work fine again. If the carpet is really wet it would be a good idea to remove it and get everything dry again as it tends to get pretty smelly if left wet. One last note: usually a leaky a/c unit will only make the passenger side carpet wet unless it's been leaking for a long time while a leaky windshield seal can make both sides wet depending on where it is leaking. Good luck, hope this helps you out!

the heater will not cause this. check under body for any damaged sheet metal. good test is go to carwash, give it a good soaking and see where it is leaking from.

there is either a torn vapor barrier on the passenger door or the sunroof drains are blocked

The drain on the a/c evaporator gets clogged, you have to raise the car on a lift, or get under the car, on the right side of the car under the fire wall you will see a black rubber hose ( about a 1/2 inch in size) pointing down, you will need to blow compressed air in the hose to clear the blockage.

It sounds like a plugged drain line from the air conditioner. Most cars now run in the a/c mode whenever u turn on the climate control (even when heat is on). The passenger side front is where it would be wet, and the carpet and carpet pad act like a sponge and will soak up the water. Let me know how u make out.