I have a s10 2000 blazer with 3 button electronic 4 wd that doesn't go into 4whi or 4wlo.When i push either of those buttons the light usually flashes once quickly and I hear a click in the kick panel of my passenger door area.Sometimes the 4w lights don't flash but I still hear a click.I don't have a service engine light or any codes when I put a code reader on the vehicle.I am looking for ideas of what to check as I do have someone mechanically inclined who can help me.I dont know the number on the transfer case,but could get it if I had to.

 

Check the DIY article - OBD2 Non-Automatic Transfer Case Diagnostics. It is exactly what you'll need to diagnose your problem seeing as you don't have the auto4wd option.