The Federal Reserve is illegal. It was ruled unconstitutional after the Civil War for one, mainly because they used fiat currency over a backed currency. But technically, the Congress has the power "To coin Money, regulate the Value thereof, and of foreign Coin, and fix the Standard of Weights and Measures;". So it's not an executive branch thing.
But trust me, there are more and better reasons to close the Fed than that. Start with how it's not even a government agency, the Federal Reserve is a private bank who gets to make the currency for the largest economic nation in the world, with little to no oversight on how much they print off. The right to coin money should be in the hands of the people, in the form of the Treasury of the US, not in the private banking cartels hands.