At one time, BATFE (and their predecessors) took a lenient view and allowed registration of NFA firearms when they came into a person's hands innocently (inheritance, for example). But under the Clinton administration, they tightened things up and now will consider any such gun (machinegun, sawed off shotgun, suppressor, etc.) to be contraband. They have been pretty decent about innocent possession, though, just accepting the turn-in.
I won't advise destroying the gun, for the reason that even that would be technically illegal. Of course, if your area boasts a deep lake or you have a boat that can go to sea....
If it is any consolation, "Buckingham" was a name used on very inexpensive shotguns that today are almost worthless, if not positively dangerous, so whatever happens it would be no great loss.
Jim