No, Stormtroopers were a specialist unit of the German Army during WWI - Trained in infiltration tactics. The Nazis commandeered the name to use for it's squad of brown shirted streetfighting bullies.
It has no original connection to the Nazis.
Just like they took the Hindu symbol for peace and applied it to become the Swastika. Now everyone associates that with Nazis when really it was the Hindus.