there are a few possibilities of what could be causing your particular issue.
(1) if your blazer is a dash shift, (2hi - 4hi - 4lo) and when you press the 4wd button and the light blinks a few times and goes back to 2wd check the fuses then wiring.
(2) if you engage the 4wd and get a whirring noise that seems to increase when you brake... there is a linear actuator on the front diff pot that commonly malfunctions on this model. these are usually available from a wreckers (I believe the interchange is 95-96) 3 bolt fairly easy fix... also check the vacuum (actuator) unit under the battery tray on the passenger side it has to be unhooked to remove the linear actuator anyhow. and make sure you unplug the little plug on the side.
(3) the transfer case may not be functioning. other than taking it to a mechanic the only thing I can think of is disconnect the front drive shaft attempt to put it in 4wd (make sure the vehicle is secure preferably turned off) and manually try to turn the front coupling that the drive shaft was attached to by hand.
(4) if you get a noise from the front left or right and grinds a lot when you turn you may need to replace the front axles (the ends can become damaged and effect the 4wd rare but apparently happens, you will more than likely need to replace the front diff as well if this is the case)
(5) if you select 4 and get a whirring and grinding or rattling/grinding from the front (or have eliminated the rest) have a look at the front diff pot (painful job :P) take your time you may have to disassemble the entire front suspension remove the front drive shaft unbolt the 4 through bolts that hold the assembly in place (you may have to unhook the Power Steering box and linkage and move it out of the way too) it will take some careful twisting and shifting to get it out but though difficult it is possible with almost no force.
these are all that come to mind right now I hope this helps if you can have a licenced mechanic or someone with the relevent mechanical experience look over the problem
A 1996 Chevy Blazer Does have a transfer case IF it is equipped with 4 wheel drive
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