“Under what legal authority can they demand the reports?”
There’s no legal basis for this requirement. None. The reporting requirements for multiple handguns is explicitly spelled out by an act of Congress. There are no other requirements for multiple purchases of any other types of weapons.
Presumably the NRA will immediately sue Obama in multiple Federal courts on behalf of hundreds of gun dealers in the affected states.
The Won hath spoken all shall heed his wise words. Now get back to work so you pay your taxes serf!