you tell me the year ill tell you why
A failing cylinder
timing is offset
Incorrect Timing Can Cause Backfire.
Poor maintenance and/or bad ignition coil.
Sounds like timing is off,you didn't brake your timing belt did you.
Before I can answer your question I need to know if the engine backfires during any other operations, such as putting the car in drive?
yes they did make a 4 seater corvette.
You need to replace the Distributor cap and rotor. Could have a bad coil are coil wire too.
Incomplete combustion
Could be in need of a tune-up.
Start and run, no. Cough, backfire, yes.
Chevrolet didn't make the Corvette in 1983.