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This may be a tough one. If your's is the same as my '96 I can tell you how to get the control head out of the dash, disassemble it and remove the bulbs. However my service manual does not have a list of the replacement bulb that I can find. This link is a summary of how I removed and disassembled my control head from the dash
According to my '89 FSM, the Air Conditioner/Heater Controls lamp is a trade #37. I'd check at a local auto parts store first. Hopefully it's a standard, low intensity bulb.
#37!
Had to go to three stores to get it. One fixed!
Got a bulb out on my 98 blazer too. Bulbs are soldered into the board on that one. I bet they are soldered in on the 96 vette too.
May not be so easy....
The climate control bulbs are not soldered in on a '96. They are inside the climate control box. After removing the control box and opening it up you need a small screw driver to turn the bulb a quarter turn to remove it from the circuit board it is mounted on.
Designer Imagines A Corvette That Looks More Like a Corvette Than the Corvette
Slideshow: A Jaguar designer's personal project imagines what a modern front-engined Corvette might look like if Chevrolet revisited the golden age of the Stingray.