Does the Vampire Knight manga have additional story that the anime doesn't? I've watched the Vampire Knight anime. I really like the story so far but find it frustrating that the anime doesn't finish off the story. I'm trying to figure out if it is worth reading the manga.

Does the manga contain more story than the anime? Does it finish the story off?
Based on Manga Updates' page of Vampire Knight:

Status in Country of Origin
19 Volumes (Complete) + 2 Bonus Chapters
10 Bunkoban Volumes (Complete)
Anime Start/End Chapter
Starts at Vol 1, Chap 1
Ends at Vol 10, Chap 46

If you wonder why there are two kinds of volume above, in Japan, usually &quot;volume&quot; as we know it called tankoubon, and bunkouban is a tankoubon with more pages, thus why bunkouban has less volume. Further reading in Wikipedia about those formats.
So, to answer your question:

Does the manga contain more story than the anime?

Should be, because the anime only caught up to vol. 10 and when all season of Vampire Knight anime ended, the manga was still ongoing. You can see it based on the year the anime and manga ended at Wikipedia's page of Vampire Knight.
Further, I didn't watch the anime and only read its manga till the end, but seeing Wikipedia's page of List of Vampire Knight episodes, seeing the story of the last episode of Vampire Knight: Guilty, I can tell you that's not how the manga ended.

I'm trying to figure out if it is worth reading the manga. Does it finish the story off?

Whether it's worth or not, it's up to each reader. But if you want to know the &quot;official&quot; ending (I said &quot;official&quot;, because the manga is the original source), then it's recommended to read the manga, because the manga did finish it off.