My 2000 zr2 blazer wont go into 4wd wheni push the button,the light just blinks,and i hear a clicking noise from behind the glovebox

Look online for diagnostic sequence to get closer to problem. This is a common problem with several fixes. I have a 2000 ZR2 with 205,000 miles in Michigan also that has had several issues. The aluminum wire from front to back has been a challenge also.

I had the same problem with my blazer. Light would just blink and not shift into any 4x4 mode. On my blazer there was a vacume hose disconnected on the center of the firewall near thhe top. I plugged it back in and I had my 4x4 back after 3 years.

Its probley transfer case trubel but check fuse and fluid level in case/jack up all 4 wells and try the butten and see if will ingage,see if both drive shafts are moveing,if so you mite have frount diff truble.
tim

If it's like any other Blazer, a possible simple fix is to disconnect the battery for about 10 minutes. That resets the computer system. It worked for my Blazer. I had the same problem. Mine went into low range, but my front axle wouldn't engage. So give it a try. The only thing you'll mess up is the clock in the radio.