Executive Orders only apply to the running of the Executive branch of government. They may not be used to circumvent constitutional protections or the authority of the Legislative branch...but somebody may have to get their heads slapped around first, just to get their attention on this.
So the kind of EO I see arising from this would ban all Executive Departments, agencies etc from doing business with any company manufacturing or selling the banned classes of weapons to the public.
Further, this ban could be cascaded down to state and local governments through the witholding of grants and the like.
The goal will be achieved by Using EOs targeting the manufacurers and sellers though trying to dry up a substantial portion of their business through removing governmental customers.