I've known this to be a false statement for decades due to the fact that I used to have in my possession (subsequently lost in a move) a copy of a 1930's issue of the American Rifleman which had in it an opinion piece about the, then current, US Attorney General pushing for gun legislation/laws that would violate the US Constitution. If I recall correctly the magazine was from the first half of the 1930's.
I just visited the current NRA history web page and found this paragraph concerning the early 20th century:
For some reason I recall seeing antis stating that the NRA became active in politics in the mid 1960's. The above would indicate that to not be true.Through the association's magazine, The American Rifleman, members were kept abreast of new firearms bills, although the lag time in publishing often prevented the necessary information from going out quickly. In response to repeated attacks on the Second Amendment rights, NRA formed the Legislative Affairs Division in 1934. While NRA did not lobby directly at this time, it did mail out legislative facts and analyses to members, whereby they could take action on their own. In 1975, recognizing the critical need for political defense of the Second Amendment, NRA formed the Institute for Legislative Action, or ILA.
For an example of an anti NRA version of history there's this:
The Suprising Unknown History of the NRA
This page is so slanted I have a hard time reading it.
And some verifiable facts:
10 Surprising Facts About the NRA That You Never Hear