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Might also be the tach filter too. It's a metal tube shaped device bolted to the back of the intake manifold. A wire should run from the distributor to the filter and then from the other end of filter to the tach.
If the filter is open, you can just connect a jumper wire across it to see if the tach starts working again. However don't leave it this way for any length of time as the filter is there to supress any spikes, etc that could damage the tach over time.
If this is the issue you can get a replacement from several corvette vendors or you can build your own using a couple of simple components available at Radio Shack, assuming you don't need the original shape/size for originality. If you need a copy of the circuit diagram, let me know.
If the tach is not working at all or reads a couple thousand rpm high... the most likely problem is the tach circuit card. The card is located on the rear of the tach head. This a simple job to replace the card. Did mine in about 15 min. Good luck. By the way, check the tach wiring harness from the distributor cap, the ground wire could be the problem.