Who actually gave Grima his nickname, 'Wormtongue'? I doubt his parents did. I would guess it would be either Saruman or the Rohirrim.
Wormtongue seems to be a nickname used by the Rohirrim - as Gandalf says to Théoden:
But first send for Éomer. Do I not guess rightly that you hold him prisoner, by the counsel of Gríma, of him that all save you name the Wormtongue?
I can't think of an occasion that Saruman calls him Wormtongue explicitly. After their defeat at Isengard he refers to him as "Worm", presumably referring both to the previous nickname and to his debased status.