Need help with vibration

For your 360 and 904 trans, you should be using the 10239 flexplate. There are multiple B&M flexplate with the bite out of them, some for the 727 and some for the 340 cast crank, so be sure you have the right one.

Since the vibration goes away when you let off the gas, it certainly does sound like a cylinder miss. After you check the park and plugs, then I'd run a compression check to see if any cylinder is way off form the others. If so, then use air in the cylinder with the valves fully closed to see where the air is leaking out.

At some point, if nothing else pops up obvious, then pull the valve covers an carefully check the lift on each and every valve, looking for one that is not lifting fully, to see if a cam lobe is wiped out.

What do you have for a carb? As a first check, look down the carb throat with a mirror while running and see if you have drops of gas dipping or dribbling down the throat.