4x4 on 89 bronco ii won't engage

You have a switch on the instrument panel ?
The NVG 263 transfer case features a three button shift control switch located on the instrument panel
The NVG 236/246 transfer case features a 4 button shift control switch located on the instrument panel.
Or
The New Venture Gear (NVG) 261, RPO NP2 transfer case is a two-speed, part-time with "mode shift-on-the-fly" capability. It has a chain driven front output shaft and an epicyclical low range planetary arrangement. The NVG 261 transfer case features a four position shift lever control located in the vehicle floor plan. As required, the operator can select 4HI position from 2HI "on-the-fly," as described in the owners manual. A dash 4WD lamp will continue flashing during shifting, until all criteria have been met and the new mode/range position has been reached. Once the new mode/range position is fully engaged and the front axle disconnect locks in, the dash light 4WD indicator lamp will remain ON constantly. Range shifting functions similarly, although it should be limited to speeds 8 km/h (5 mph) or less.

there is a 4X4 controller--top of right side dash--will have to be checked at that point.

ON LEVEL GROUND
With the transmission selector in "park", start the motor.
With your foot on the brake pedal (steady-constant pressure), move the transmission selector gear to neutral- then turn the 4 X 4 selector knob to 4-wheel drive high. Wait for the 4 X 4 engagement.
Once it is in 4-wheel drive high, with the transmission selector still in neutral, quickly pump the brake peddle twice, keeping you foot on the brake pedal on the second pump, turn the 4 X 4 selector knob from high range to low range. The transfer case should then change from 4-wheel drive high to low.
Electronics being what they are, you sometimes have to repeat the process. You may even have to inch forward or backward.
[I have an '01 Excursion and an '05 F250 with 4 X 4 on the fly...I miss my '97 F250 with the manual 4 X 4 shift on-the-floor!]
Anyway, check the fuse panel for any blown fuses.
Double check to make sure that the problem is not just blown 4 X 4 lights on the instrument cluster. With the nob turned to 4-wheel drive, have a second person- a spotter, check to see if the drive shaft to the transfer case is moving. If it isn't, you have to delve into the troubleshooting process.
Check the solenoids, vacuum hoses, vacuum actuator, and transmission range sensor on the transfer case. .

re/check your oil level as it possibly lower then you thought ,as an auto gearbox needs the level to be full, dont rev the gearbox trying to get it to move this will damage the bands inside the gear box,

Engage the 4wd and go underneath and tap the 4x4 motor to engage it, it is probably just stuck. More than likely its not a fuse.

Is the system ESOF ( Electronic Shift On the Fly ) or MSOF ( Mechanical Shift On the Fly ) ? Lift and support the front of the truck so the front wheels are off the ground ( take safety precautions, chock wheels, jack stands, etc ). Key on, engage 4x4 and try to turn the front drive shaft. If it turns it is a problem with the transfer case. If it doesn't it is a front axle engagement problem.

sounds like the shift motor on the transfer case is not working

I had the same problem with the shift on the fly push-button 4x4 on my 97 Suburban. GM Shop manual indicates that it could be due to an encoder error on the actuator motor. Try pulling the transfer case fuse to reset the transfer case control module. Wait at least one minute. Re-install fuse then re-try operation. Did it last week and that saved me a trip to the dealership and extensive troubleshooting.

The sound is not coming from the glove compartment, it is coming from the panel behind the drivers seat, that's where the relays and computer are located.
The relays click and apply power to a transfer case motor.
The motor attempts to turn a shaft that engages the front drive shaft.
When the computer thinks the shaft has NOT turned it tries again.
It will try three times then quit, figuring the shaft has not turned.

Get someone to crawl under the vehicle and engage the 4WD (just turn on the ignition- do not start the engine) See if they can hear the motor attempt to turn the shaft, it will make a clicking sound.
If not the connections or the motor may be bad.

I have found the most probable cause is the relays on the control board.
If you are technically adept --
you can remove the small side panel from behind the drivers seat and remove the control board. (remove the power plugs first)
Clean the relay contacts and bend them a little to give them more 'wipe' to keep them clean longer.
Put it back in and turn on the ignition.
Then push the RESET button on the side of the board.
Try the in / out 4WD a couple of times. If it doesn't work turn off the ignition, turn it back on and push the reset button again.

Check to see if you are getting power through your switch on your dash. I also suspect your shift motor but it is a lot easier to check it with a test light.