When referencing story, I'd assume they mean the Main Quest for the award.
Could be wrong about that, but everything outside of the Main Quest is just add-on content in my opinion.
Judging from IAA's description of their award ( http://www.interactive.org/awards/2011_14th_awards.asp), I would say that the the Outstanding Achievement in Story is not tied to main quests:
Presented to the individual or team whose work has furthered the interactive experience through the creation of a game world -- whether an original creation, one adapted from previously existing material or an extension of an existing property which best exemplifies the coalescence of setting, characters and plot.
A Khajiit merchant handing me a thank-you gift, a giant continually pounding already-dead enemies near the body of his dead mammoth, and thugs showing up to pay me back for thievery, are all embodiments of story. Exploring, and finding and reading journals, to unravel the fates of expeditions and of families, constitutes story. Ad-lib adventures, just wandering into some cave and picking fights with wolves or picking people's pockets, constitutes story.
Quests, and Radiant Story, and all our ad-lib doings effect story. Setting -- visual, audial, political, and elsewise -- effects story. Characters, which include everything from you, the central figure of the game, to the lowliest rabbit, effect story. Point of view effects story. (Please note that I really do mean "effect", not "affect." Thank you

).
Everyone who plays Skyrim knows that has stories to tell. Not so many of us recognize Skyrim itself as one, big interactive story.