The Click sound is from the 4x4 module behind the radio, the sound usually means it is working, but not always
Could be shift motor fuse is blown in engine fuse box, check Fuse #27, 20amp it powers the shift motor
And it could be the shift motor, get under the truck and tap on the Motor part, the brushes inside can get stuck so no power to the motor, if it starts to work again you will need to rebuild or replace it, as it will quit again
The way it works
Switch on the dash is just 3 resistors and NOT connected to the 4WD lights
4x4 module powers up with key on
It sends 5 volts to switch, it gets back 1 to 4 volts depending on switch position
The shift motor has a contact wheel and 4 contacts, each contact is a position, 2WD, 4High, neutral, or 4Low
4x4 module checks contact position, if it matches switch selection then nothing happens
When you select 4high, the voltage from the switch to the 4x4 module changes
The 4x4 module has 2 Relays inside, it closes 1 relay to power up and turn shift motor clockwise, and the other relay is used to turn it counter clockwise
shift motor turns to 4high position known by that contact on the motor closing
4x4 module THEN turns on the 4high light on the dash
If 4WD lights flash it means something is wrong, shift motor is not moving or contacts are not changing
You don't mention flashing 4WD lights, so the 4x4 module may be at fault
The 2001 to 2005 4x4 modules were a known issue, so check the fuse and shift motor first, but it may need a new 4x4 module
Shift motor guts seen here:
https://www.explorerforum.com/forums/index.php?threads/how-to-rebuild-4x4-shift-motor.123542/