What is wrong with it? Typically what breaks on the 84-87 motors are the plastic gears inside of the motor. There are 2 of them. A larger white one, and a smaller black one. You can buy replacement gears from Ecklers for about $45 (I think thats the price), and repair them yourself if you want to.
Re: Headlight motor repair on 1986 Vette (scvettedriver)
It is actually very easy to fix your headlight if it is the gear. If the motor runs but the light doesn't move, you need new gears. If the motor doesn't run, it could be either the motor or the electrical connection.
Disassembly is very easy, just a few nuts and screws...don't really need a manual. You'll probably have to pry the case apart after you remove all the screws. There is a large and a small plastic gear inside. I replaced mine with a similar set of plastic ones from Mid-America and it worked fine for a couple of weeks until my motor (which needs cleaned) didn't close it and I tried to 'help' it. After stripping my new gear :( I had to replace it again. The brass gears are more expensive, but they will last.
GM parts quoted me $1100 for 2 complete headlight assemblies