First as a sanity check, I would check all fuses. There should be a fuse diagram in your owner's manual, in particular there should be a larger fuse, "maybe" in the under hood power distribution box, that goes to the shift control module.
There may be another going to the accessory delay circuit that's also needed, and then on the interior fuse panel, one going to the battery saver relay. I'd just take a multimeter and probe into the back of all fuses to check continuity.
Backing up a bit you could check from the other end, whether there is 12V on the brown wire going to the transfer case clutch coil.
While I'm not aware of anything involving the instrument cluster itself that would need plugged in, there is probably a torque on demand relay (solid state, does not look like other relays in the vehicle) roughly behind the radio. This location is only a guess, they might've moved some things around in an '02 Sport compared to earlier Explorers.
I take that back, is it possible that your 4WD is working but it's just that the light for it isn't? The power for that light seems to come from the instrument illumination circuit but I don't know where it connects physically.
I'd also measure resistance across the 4WD bulb to make sure it's not burnt out, or if they had switched to an LED then "some" meters can check those with a diode test mode.
Anyway if you have 12V going to the transfer case clutch coil on the brown wire and you're sure it's not engaging 4WD then it's a problem in the transfer case, but from what I can tell the 4WD light should be on, assuming not burnt out, assuming there is only ONE thing wrong.
The vehicle is now 16 years old, it could be something unrelated to the dash like just that the 4WD switch has dirty contacts or has failed. Again I don't know if it is the same setup as the past generation Explorers but if it is (and these wire colors Might Be Wrong) then you should have a white/light-blue wiring coming from the GEM and a gray/red wire going back to the GEM.
Changing the switch position should vary the resistance across the switch from maybe 3.9K Ohms in one position to 1.1K to 360 Ohms in the 3rd position... assuming 3 switch positions.
Here is a wiring diagram for a 2nd gen Explorer 4WD setup. I am not certain how much yours might have changed - or it might be identical.