No they couldn't because A) the gold doesn't belong to them, and B) the Fed doesn't deal with individuals. Around 98% of the gold in the vault at the New York Fed belongs to foreign countries. Virtually all the rest belongs to international agencies. Only a fraction of a percent belongs to the U.S.
It wouldn't be the first time bankers started playing fast and lose with valuables in their trust. Not allowed is a long way away from can't be done, especially with an organization which is never audited.
and B) the Fed doesn't deal with individuals.
Then replace persons A through D in my example with major banks and investment companies which already deal with the Fed in billion dollar amounts. What happens when Chase or Bank of America comes up with a "great idea" where the Fed can pick up 1 or 2% of the value of their gold with absolutely no risk and the gold doesn't even have to leave their vault? Everything is fine just so long as no country asks for a physical withdrawal of the gold.