As far as I can tell - not an expert on this - the statutory authority for the president's executive actions on background checks stems from this law, as well as its follow up in 1938. The 1934 law places a tax - an excise tax - on the sale of machine guns and sawed off shotguns, meaning both were still legal. This was during a time when Congress was expanding - and the Supreme Court was accepting - the idea that the tax and spending clause of the Constitution allowed taxes to be used to as a regulatory device, not just to collect revenue for predefined purposes.
Since it was a tax bill, it was originally part of the Internal Revenue Code.
The 1934 law also required all sales to be recorded in a national registry. The 1938 act expanded the law to require the licensing of interstate gun dealers, mandated that they record their sales, and prohibited sales of guns to to people convicted or indicted of committing violent crimes. Later laws modify these requirements - as do the recent executive actions - but this seem to be the basis of statutory authority.
Why did it take until 1934 for the national government to get into the business of regulating guns? Prior to this time, regulation was a fully state affair, but the availability of cars and better roads made it easier for criminals to evade states. This explains the expansion of the federal government into criminal law in general.
Here's a bit more on the original act:
- The 1934 National Firearms Act, and why it is obsolete.
- Franklin Roosevelt: The Father of Gun Control.
- Fully-Automatic Firearms.
- Full text of the law.
- 26 U.S. Code § 5845.
