Does Gríma have a surname? Wormtongue was a nickname/epithet and not his actual name - but did he have a genuine one?
In The Lord of the Rings, Hobbits of the Shire and Bree and the Men of Bree are seen to have surnames...
The Men of Bree seemed all to have rather botanical (and to the Shire-folk rather odd) names, like Rushlight, Goatleaf, Heathertoes, Appledore, Thistlewool and Ferny (not to mention Butterbur). Some of the hobbits had similar names. The Mugworts, for instance, seemed numerous. But most of them had natural names, such as Banks, Brockhouse, Longholes, Sandheaver, and Tunnelly, many of which were used in the Shire.
...but not everyone in the wide world of Middle-earth have them.
What you're looking at here in the name "Wormtongue" is an epithet, or as it is also called, a nickname. Most people lack a surname, with their names ending with "son of ..." instead. Eg. Aragorn son of Arathorn, Gimli son of Gloin.
In Grima's case, he would just be addressed as Grima, son of Galmod, of Rohan, and therefore does not have a surname.
To paraphrase from an excellent answer found here; it would seem that the use of surnames is almost uniquely found in the Shire and Bree. Hobbits, who love genealogy and the study of their family trees and such, used surnames to distinguish themselves and others based on their family. Bree, a town "heavily influenced by hobbit culture" (by having that peculiar Hobbit-Men relationship), started using surnames as well. It would appear that this culture did not spread to lands elsewhere.