A central issue in Mrs. Clinton’s proposals are the background checks on prospective gun buyers, which are required for retailers at stores. But under federal law, they are not required at gun shows or over the Internet with private sellers.

Under Mrs. Clinton’s plan, she would use administrative powers to make anyone selling a substantial number of guns declared “in the business” of firearms dealing, and subject to the same rules as retailers, if Congress does not act, according to the campaign aide.

It was not immediately clear what the bar for being declared “in the business” would be. And use of executive action in connection with guns is certain to face criticism from staunch supporters of the Second Amendment. It is also likely to be applauded by Democrats who have grown weary of gridlock in Congress.