I have long supposed that the origin lies in NAZI Germany’s special weapon in World War II:
The Sturmgewehr (”Assault Rifle”) 44 [StG 44], a.k.a. Maschinenpistole 43 [MP 43] or Maschinenpistole 44 [MP 44].
It was a selective-fire weapon and, I opine, may have been the inspiration for the AK-47.
For years I’ve heard that the term assault weapon or rifle came from the Nazi military command. They were trying to convince Hitler to approve the manufacture of a new select-fire weapon and pitched it as an “assault rifle” because they rightfully decided that Hitler would like the name.
I don’t remember where I first read that, or what weapon the writer claimed was the one the generals were pitching. Over the years I have heard and read that story several times basically unchanged and always the emphasis was that the term was coined by Nazi generals to sell Hitler on authorizing manufacturing a gun they wanted.