In my head cannon, much like how the hobbit is written as a narrator telling a tale from the future, I assume Grima earned the moniker of worm tongue.

My impression was always that 'Wormtongue' was a derogatory nickname given to Grima by those who secretly despised him and were truly loyal to Rohan and Theoden. I never felt that it was his actual name.

As other comments have pointed out, the name "Wormtongue" was just a nickname that those in Edoras called Gríma son of Gálmód. The name just officially stuck once Gandalf the White strolled on in and showed Théoden that Gríma was indeed a wormtongue

let's be clear, any society that believes nomonative determinism would be considered a bit shitty.