Ignition buzzer. Likely the ignition switch is worn and it is buzzing at the incorrect position. If the doors are off or open, it will trigger with the key in the ignition. (but if it is doing so with the key out, and stops when you wiggle the switch "tabs", then it is likely worn and not working correctly.) I disconnected mine on my '68, because it bothers me to hear it.
Thanks for all the input everyone! It appears it's the ignition buzzer but I did have a hard time identifying these pink wires that I have unplugged at the moment. Another member suggested they might be the speed sensor wires. Where do they plug into and would them not being plugged in cause these issues?
Do you know what a speed alert speedometer looks like? If the car is not equipped with one, it still has the wires and plug, just not connected to anything.
I can't seem to find one for a non-tilt column shift column. You wouldn't happen to know where I could buy one, do you?
I searched on google and eBay and couldn't find one. Didn't think to search the site you sent. I should have thought of that but I didn't. My mistake. Appreciate your help anyway.
Okay now that things are apart and I have the new part it doesn't seem like the buzzer switch is the problem. It looks like it's the ignition cylinder tab that hits the buzzer. It's not retracting when I remove the key.
You might be able to remove the tumbler, disassemble, clean and adjust, then put it back together. (I have done many repairs on lock mechanisms, but not specifically one like yours, so I cannot guide you on the details) Call around and talk with locksmiths, the "older guys" may have the knowledge on how to repair it. It my just need a good cleaning and proper dry lubricant (I use Dow Corning Molykote D-321R).