Radio not working

The buzz is most likely from the alternator. That's why GM put all that ignition shielding on the cars...and all of those danged capacitors on every electrical component.
You can get a capacitor to install on the alternator to solve the problem.
You could also re-route the electrical connections for the stereo directly to the battery, and eliminate 90% of the noise, but that's a pain too.
The antenna ground plane is under the body panel on the rear, where the antenna is. The antenna goes thru this piece of metal, usually aluminuminum, and then there is a strap screwed to that that goes to the frame, and provides a solid ground plane for the antenna itself. I ain't no radio-wave specialist, but any antenna must have some sort of ground plane in order to receive a clear radio signal from the air. On "regular" vehicles, the metal bodies provide the ground plane, but on Corvettes, there is no metal body, so we have to have the small metal plate for the antenna.
Normally, if the antenna doesn't have a good ground plane, the radio just doesn't receive very well, but it usually won't create a buzzing noise. But, since this is Adam's car....
You can crawl under the rear of the car where the antenna is and see the plate, and then look to see if there is a strap attached to it that goes to the frame.