This can be caused by a few components.
When you push the button can you hear the transfer case shift?
If no you have either a transfer case control module or encoder motor problem.
If yes you need to see if the actuator for the front axle is engaging. The diagram is the vacuum control for the front axle. Vacuum source is a port on the engine. When the transfer case shifts to any four wheel drive position the "switch" connects the blue and red lines and pulls the actuator in which moves a collar in the front axle to make it drive both wheels.
To see where the issue is you need to see if the actuator (other picture) is working. It is mounted on the passenger inner fender near or under the battery. The circled clip likes to rust or pop out of the cable as well so check to be sure the cable is actually connected. The rubber diaphragms dry rot and crack so they will not pull in or battery acid drips on them causing them to fail. You should be able to see that type of damage. If it looks okay you can test it, apply a vacuum to the port that the hose connects to, the diaphragm should pull in and pull the cable, and stay retracted until you remove the vacuum.
If it tests okay it is likely that the vacuum switch on the transfer case or one of the vacuum lines has failed.
The most common problems are the T-Case motor, the vacuum switch and the actuator. The control module and switch are the next ones (yours sounds okay, normally the four wheel drive lights will flash or you push the button and only the two wheel drive light stays on).
Images (Click to enlarge)
Sunday, January 8th, 2017 AT 7:58 AM