This is really interesting. I never noticed that some issues of Weird Tales had an NRA logo. So I looked it up and I ran across this blog post where a woman had done the same thing a few years ago.
Long story short, it looks like the NRA being represented on Weird Tales is the National Recovery Administration, not the rifle people. This other NRA was an organization created during FDR's administration to help facilitate negotiations between workers and employers and to set prices and fair practices, and so forth. Ostensibly, an organization meant to mitigate the worst instincts of capitalism with relation to workers and bosses. Not exactly the kind of thing the rifle people are interested in.
This cover is from 1934. The NRA (not the gun one) was created in 1933. I don't know much more about it, I just found it very interesting. One of those "Well HUH" moments for me since I do love looking at Weird Tales covers but I never paid attention to this.