Broad authority to regulate firearms was given to the Executive branch by the National Firearms Act of 1934 and the Gun Control Act of 1968. Under these statutes the Fed established the BATFE, which issues ongoing rulings about administrative questions as they arise, such as "Does this [insert device name here] constitute an NFA weapon under Federal law?"
I suspect that the President does have broad authority under these acts to provide direction to BATFE about how he wants the laws administered, such as (for example) putting all weapons that are cosmetically similar to military rifles on the NFA list, making them subject to federal registration and a $200 transfer tax.
So yeah, I think he can do this. More importantly, HE thinks he can do this. Whether he's right or not will be for the courts to decide, most likely several years from now.
Like I said he can try but its going to be his undoing. Obama is no different from Stalin or Mao. They think there is no line in the sand. Guns are the line in the sand here.