1999 Chevy Blazer transfercase prob.

If you have no lights on the switch I would start there. The switch modules do have a problem with poor solder joints.

A common problem for Chevy Blazer 4x4 is for the activation switch to fail, or the transfer case shift motor/sensor to fail. The last guy is right, start with switch and then move on to the shift motor.
Note: The motor failure will usually trip a code.

You will need: small screwdriver, q-tips, and rubbing alcohol.
Pop out the 4WD button assembly from the dash. Unplug the the two wire connectors. Hold the button assembly in your hand face down. make sure you are in a place that, if you were to drop something small you can find it easily, but lets just hope you're careful. With a small screwdriver, pry the back of the assembly from the front....remember to keep the face down in your hands at all times. You see those blue boots with metal contacts on them? These are the things that will fall out easily. Take your time, remember where these boots are located they will pop out easily. In fact , have someone handy with a camera to take a picture of it before going further. Now with a q-tip dipped in alcohol gently clean the metal contacts in the blue boots. Then, with the q-tip and alcohol, clean the whole face of the board that the blue boots come in contact with, paying close attention to the white circle patterns on the board. They may not look dirty but it doesnt take much to make this assembly act nutty. When done cleaning and parts a fully dry, re-assemble and give it a try.