F150 4wd will not engage

Manual shift or Four-Wheel Drive Systems - Electronic Shift I'm assuming you have electronic shift because you refer to relay clicking behind the instrument panel . Principles of Operation Transfer Case - Electronic Shift
The four-wheel drive, electronic shift-on-the-fly feature electrically shifts the vehicle transfer case between 2WD, 4X4 HIGH, and 4X4 LOW. The system mode is selected by the operator through the mode select switch (MSS) on the instrument panel. The operator is informed which mode the system is in by two instrument cluster indicators, one for 4X4 HIGH which is 4X4, and one for 4X4 LOW which is LOW RANGE (in 4X4 LOW both the lamps are on). Shifts into 4X4 HIGH can be made at any speed. When shifting into 4X4 HIGH with the vehicle stationary, tooth blockage may occur, preventing shift completion. When the vehicle is driven above 8 km/h (5mph) the shift will complete. When shifting in or out of 4X4 LOW, the generic electronic module (GEM) requires that the vehicle speed be less than 5 km/h (3 mph), the brake pedal be applied, and the transmission in NEUTRAL (automatic transmission) or the clutch pedal be depressed (manual transmission). (The digital transmission range [TR] sensor informs the GEM when the automatic transmission is in the NEUTRAL range position.)
The gearmotor encoder assembly is mounted externally on the transfer case. It drives a rotary cam which moves the mode fork and the range fork within the transfer case between the 4X4 HIGH, 4X4 LOW, and 2WD range positions.
The GEM uses two relays to control the gearmotor encoder assembly shift between 4X4 HIGH, 4X4 LOW, and 2WD modes. The GEM controls the pulse vacuum hublock (PVH) solenoid which supplies engagement and disengagement vacuum to control the hublocks.
The GEM accomplishes shifts by interpreting inputs from:

• mode select switch (MSS)
• vehicle speed signal (transmitted from the ABS system)
• gearmotor encoder plate position
• brake pedal switch
• digital transmission range (TR) sensor (automatic transmission)
• clutch pedal position (CPP) switch (manual transmission)
• ignition switch

Based on these inputs, the GEM controls the shifts into 2WD, 4X4 HIGH, or 4X4 LOW with the following outputs:

• low to high relay (clockwise)
• high to low relay (counterclockwise)
• pulse vacuum hublock (PVH) solenoid
• gearmotor encoder assembly

The electronic shift-on-the-fly (ESOF) system has a feature which allows the driver to override the vacuum operated hublocks. When the front hublocks are manually turned to the LOCK position, the hublocks are locked at all times, overriding the vacuum operated system. If the front hublocks are manually turned to the AUTO position, the hublocks can only be locked by turning the MSS to 4X4 HIGH or 4X4 LOW position.
Do you know what DTC'S - diagnostic trouble code's are ?
GEM Diagnostic Trouble Code (DTC) Index
GEM Diagnostic Trouble Code (DTC) Index DTC Description Source Action C1728 Transfer Case Unable to Transition Between 2H and 4H GEM CARRY OUT the electronic shift function test. REFER to Function Test - Electronic Shift in this section. C1729 Transfer Case Unable to Transition Between 4H and 4L GEM CARRY OUT the electronic shift function test. REFER to Function Test - Electronic Shift in this section. P0500 Vehicle Speed Sensor (VSS) Malfunction GEM GO to Pinpoint Test F . P1804 Transmission 4-Wheel Drive High Indicator Circuit Failure GEM GO to Pinpoint Test H . P1808 Transmission 4-Wheel Drive Low Indicator Circuit Failure GEM GO to Pinpoint Test I . P1812 Transmission 4-Wheel Drive Mode Select Circuit Failure GEM GO to Pinpoint Test E . P1815 Transmission 4-Wheel Drive Mode Select Short Circuit to Ground GEM GO to Pinpoint Test E . P1819 Transmission Park/Neutral Position Switch Short Circuit to Ground GEM GO to Pinpoint Test F . P1820 Transmission Transfer Case Clockwise Shift Relay Coil Circuit Failure GEM GO to Pinpoint Test E . P1822 Transmission Transfer Case Clockwise Shift Relay Coil Short to Battery GEM GO to Pinpoint Test E . P1828 Transfer Case Counterclockwise Shift Relay Coil Circuit Failure GEM GO to Pinpoint Test E . P1830 Transmission Transfer Case Counterclockwise Shift Relay Coil Short Circuit to Battery GEM GO to Pinpoint Test E . P1832 Transmission Transfer Case Differential Lockup Solenoid Circuit Failure GEM GO to Pinpoint Test G . P1834 Transmission Transfer Case Differential Lockup Solenoid Short Circuit to Battery GEM GO to Pinpoint Test G . P1865 Transmission Transfer Case Contact Plate Power Short to Ground GEM GO to Pinpoint Test E . P1866 Transmission Transfer Case System Concern - Servicing Required GEM CARRY OUT the electronic shift function test. REFER to Function Test - Electronic Shift in this section. P1867 Transmission Transfer Case Contact Plate General Circuit Failure GEM GO to Pinpoint Test E . P1876 Transmission Transfer Case 2-Wheel Drive Solenoid Circuit Failure GEM GO to Pinpoint Test G . P1877 Transmission Transfer Case 2-Wheel Drive Solenoid Circuit Short to Battery GEM GO to Pinpoint Test G .
Your best bet , take it to a qualified repair shop .