How come Shelob's sting was able to pierce through the Mithril armor worn by Frodo and poison him?
Shelob stung Frodo in his neck, which was not covered by the mithril shirt that he wore.
As he had run forward, eager, rejoicing to be free, Shelob with hideous speed had come behind and with one swift stroke had stung him in the neck. He lay now pale, and heard no voice, and did not move.
The Two Towers: The Choices of Master Samwise