The purported concern was with "gangster weapons" but as usual, that was a smokescreen. Homer Cummings, FDR's first Attorney General, hated guns and those who owned them for any purpose. In FDR's first term, the St. Valentine's Day Massacre in Chicago gave him the excuse he was waiting for, and legislation was immediately sent to Congress.
The rationale was that all guns either had or would at some time move in interstate commerce and that the government could use its authority in that area plus its taxing authority to control the gun traffic.
The administration bill, backed by FDR, imposed federal registration and transfer taxes on guns and ammunition and also required licensing of firearms dealers who would also have to pay a license fee (another tax). For a full understanding, remember that in 1934, a new Ford sedan cost $400.
IIRC, the tax on a machinegun was to be $5000, a handgun $1000, a rifle $500, a repeating shotgun $200 and a single or double shotgun $100. Ammunition was taxed at $10 each for a handgun cartridge, $5 for a rifle cartridge, $1 for a shotgun shell, and $.50 for a .22 cartridge.
After the bill got through Congress, it required registration and a transfer tax, but only for machineguns and weapons considered "sneaky" and "un-American", like disguised guns and silencers (suppressors as folks insist on calling them now), and the requirement for dealers in those things to be licensed.
That bill, the National Firearms Act, passed in 1934. The administration claimed that its great gun control law had ended the gangster era. In fact, another administration action, initiating repeal of Prohibition did that; the NFA had little to do with it.
In 1938, re-elected to a second term, FDR went back for another bite of the gun control apple, proposing the Federal Firearms Act. This time, he again used the interstate commerce clause and taxation to propose control over those guns that Congress had refused to include in the NFA. No tax was to be imposed on those guns (rifles, shotguns, and ordinary handguns) and the registration proposal went nowhere, but the licensed dealer system was expanded to cover all firearms, with separate categories for NFA firearms and FFA firearms. That meant two different sets of definitions and different legislative language, something that Congress tried to resolve, with only partial success, when the two laws were combined (as Title 1 (FFA) and Title II (NFA) in the Gun Control Act of 1968.
Note the idea of using the federal taxation authority as a means of control, something that would be done repeatedly thereafter, and the basis for all the volumes of federal licensing laws on the books today. Also note that the FDR administration, like the Obama administration, drew up the legislation first, and then waited for a convenient time to introduce it "in response to the tragedy."
One small point. As part of the administration propaganda for the FFA, FDR, whose legs were totally paralyzed, thought he condition was kept a secret from the American people, had himself taken to the Marine rifle range at Quantico, and laid on a mat. While the fawning press corps took pictures, he pretended to fire a National Match rifle, then flashed his famous smile. This, of course, was done to convince hunters and gun owners that FDR was their friend and was not out to ban guns. (They mostly forgot about those proposed taxes on rifles and ammunition four years earlier - never underestimate the stupidity of the American voter!)
If all this sounds very familiar, it is. Just a different president wanting to disarm the American people so he can ignore the Constitution and impose his rule.
Jim