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My first thought would be the fuse. On my 69 it is the second fuse up on the right hand row, just above the short fuse. It looks like that fuse also connects the hazard flashers, do they work?
The tail lights are on a totally separate circuit than the brake lights. Otherwise the brake switch or wires on the brake pedal assembly.
The brake light switch mounts on the brake pedal arm under the driver's dash. You can use a voltmeter to see if you are getting 12vdc out of the switch when the pedal is depressed. The switch could be bad, the switch may have come loose and need to be repositioned, or you may have some other electrical problem with the brake signal wiring. The switch is the most likely problem.
The brake light signal goes through the turn indicator switch in the column. IF you are NOT turning, the signal goes to both sides, if you are turning it goes to the side opposite the turn. Do your indicators work? Check the Harmonica connector into the column. Check the fuses as well.
If you are a contortionist you can reach up under the dash and replace the brakelight switch. I couldn't do it. I have read round these parts that it helps to remove the drivers seat. You still need to be a contortionist. All this assumes that you have a replacement for your 68 only corvette only 1 year only braklight switch. Alternatively, you can get a standard $7.99 brakelight switch and with a little help from bubba you will have working brake lights.
I was able to replace mine and I wasn't smart enough to think of removing the seat. It was definitely a PITA!
If I remember correctly there are two studs mounted on the pedal assembly that the switch mounts to but it will be obvious whether I'm right or not once you get started.
The brake switch was without a doubt an improvement in the '69 model.