I read up on this issue. Mine is not engaging.  Heres what I know: when key on, engine not running,  shift to 4X4 high and listening, the pump seems to run 1-2 seconds, the 4X4 green light on the dash comes on, stays on, no 4X4.
What is the best next guess?  Actuator in the differential case stuck?  It seems it is not leaking? Access to it through any plug? Can it be freed up?
I can get it up on a lift.
Newbee here and I looked and answered a bunch of my stupid questions but now I am stumped.
TIA

How do you know it's not engaging?  The purpose of the green 4WD light is to tell you it's engaged, so what makes you think the light is on and 4WD is not engaged?

I'm not saying that you are wrong, as it is possible for the system to fail, but, it rarely tells you it's engaged when it's not - it's a lot more common for you to select 4WD and have it not engage and the light not turn on.

Tells us what you're seeing and we can walk you through a series of tests to determine where the failure lies if there is in fact a failure.

How do you know it's not engaging?  The purpose of the green 4WD light is to tell you it's engaged, so what makes you think the light is on and 4WD is not engaged?

I'm not saying that you are wrong, as it is possible for the system to fail, but, it rarely tells you it's engaged when it's not - it's a lot more common for you to select 4WD and have it not engage and the light not turn on.

Tells us what you're seeing and we can walk you through a series of tests to determine where the failure lies if there is in fact a failure.

Thanks for the reply. There is snow here in NH and the rear tire/s will spin but neither front will.

You're going to need to get under the car, at least partly for these tests, so try to find a dry place or at least a mat or similar to lie on.

Put the car in park, engine off, hand brake on, chock the rear wheels, jack one front wheel off the ground, preferably the right one (because this gives more clearance to reach the drive shaft) and secure the vehicle with jack stands. If it's a manual transmission, put it in either first or reverse, and select 2H on the transfer case shifter.

Reach under the car, grab the front drive shaft (from the transfer case to the differential) and try to rotate it - it should turn, possibly with some resistance - now select 4H, and again try to rotate the drive shaft - this time it should not rotate - it may turn slightly before locking in, but it should lock within less than one quarter turn, and you should not be able to turn it after that - if this does not happen, there is a problem in the transfer case itself.

If the drive shaft does not rotate, let's move on to the front differential free-wheel clutch.

Select 2H on the transfer case shifter and then very carefully, switch the ignition on, but do not start the engine - now select 4H, you should hear the pump run for 1~2 seconds and the green 4WD light should come on - next try to rotate the wheel that is off the ground - you should be unable to, as before it may rotate slightly before locking in, but it should lock and not rotate - if it rotates, there is a problem with the freewheel clutch setup.

Look under the front differential, you should see a rubber hose linking the axle assembly to a pipe running along the crossmember - follow the pipe back along the crossmember and forward along the chassis rail - it connects to the freewheel pump module - disconnect it from the module - if the ignition is still on, the pump should turn on and run for 10 seconds before turning off - this would indicate a working pump module and that the fault lies in either the plumbing or the differential clutch assembly - you can determine which by disconnecting the rubber hose under the diff.

Here's how the system works...

When you select 4H or 4L, a selector hub moves to engage the front output shaft and a switch on the transfer case sends a signal to the 4WD controller (on a 2000, this is most likely to be in the PCU) - at this point, the front output shaft should be turning (assuming the engine is running and the vehicle in gear), and the front output shaft will turn the front differential via the front drive shaft.  Once the 4WD controller sees that 4WD has been selected, it turns on the pump, which pressurizes the freewheel clutch actuator in the differential and engages the freewheel mechanism, supplying power to the front wheels.  Assuming that the actuator is airtight, the pump will build pressure in the system (about 6~9 psi) and this causes the pump to switch off and send a signal back to the controller that turns the light on.  If there is a leak in the system, the light will either flash or not turn on.

It is possible - if the clutch mechanism is stuck - for the actuator to build pressure and turn the light on, with out engaging the freewheel clutch, but it is not very common, and this is where the problem lies it will mean pulling the front differential assembly and opening it.

Thanks for the step by step diagnosis.  I’ll just put the car on my vehicle lift if that is OK and have my son inside the car to do the various shifts, etc. I will get back to you tomorrow

Having all four wheels off the ground changes what you will see.

Make sure the handbrake is set to stop the rear wheels from rotating and that will hold the entire drive line still during the tests - when you come to the test where you are rotating the front tire, the front tire will rotate because the other front tire is free to turn - however it should turn in the direction opposite to the one you are turning.

Once I got it up on the lift it was easy to diagnose.  The right wheel would turn but the axle did not!!  I pulled it apart and found the inner axle had lost its cir-clip and backed away from the CV joint and had shredded the end of the inner axle and the inside of the CV spline.  It would rotate freely and catch occasionally.  I have picked up a rebuilt right half shaft from O’Reily’s auto parts and will assemble it tomorrow.
I believe the rest of the 4WD looks and acts as it should.
Thanks for the help.
It turned out to be a $65 repair.  Not bad!

Has anyone found a reliable source for 4wd actuator air pumps?

What's wrong with your air pump?  They are usually quite easy to repair.

It's on a 2001 2.0 Tracker. I'm just getting a slow flashing light, no 4wd.  I separated the metal and rubber lines on the differential end and wasn't getting any air.  After some checking, figured out that the line was blocked with corrosion for about 3-4 inches. I cut that out and replaced to get air flow, still no 4wd.  I ran tubing from air pump straight to differential to bypass the restricted factory line (just in case) and still no 4wd.  Next step is to try using a "squeeze bulb" attached to differential to test the internal actuator. Hoping it's the pump.

First question - how do you know you have no 4WD?
Second question - have you considered checking for air flow directly at the pump?

Take the pump off the vehicle, there are two bolts that hold it, two rubber hoses and a three wire electrical connector - the three wires should be one each, red, black & pink, connect a 12V power source (you can use the vehicle battery) to the red & black wires, the pump should run and you should get air from one of the two hose connectors.

Quick & easy - it either works or it doesn't - and if it doesn't, it can be disassembled and in many cases repaired.

No 4wd?-I got stuck in the creek.
The pump does kick on sometimes. When it kicks on there is air flow at the pump.

No 4wd?-I got stuck in the creek.

How does you getting stuck in the creek PROVE you have no 4WD?  You could be stuck simply because of a lack of traction - you only need to loose traction on front & one rear wheel at the same time.

The pump does kick on sometimes. When it kicks on there is air flow at the pump.

If the pump does not turn on EVERY time you apply power to the red & black wires, then there is a problem with it - you can gently pry the square cap off and check where the wires are soldered to the board, see if they are loose.

I jacked front end up, locked it in 4wd and manually turned the front wheels, and only one wheel turned. 
I tested the air pump with a regulator last night, getting a steady 6 PSI.