In support of DQdlM's answer, Shelob doesn't pierce Frodo's mithril chainmail at all.
While in the book she stings him in his unprotected neck, it's clear that this doesn't happen in the movie! We see Shelob stinging Frodo from the front (since he turns to face her just before getting stung) and we can clearly see his neck is unhurt, so that's not the place. (As a bonus, we can see Shelob has a stinger, unlike real-world spiders. But that's neither here nor there).
However, I can still provide evidence that Frodo doesn't get stung through his chainmail. In this scene just minutes before his encounter with Shelob, we can see the mithril coat is actually way below his neck. Frodo's upper chest is unprotected!
Later we can clearly see at least two of his wounds: the wound from the Nazgûl, and the other must be Shelob's attack, within his exposed upper chest.
Which makes us wonder why Frodo didn't wear his chainmail in a more practical fashion.
Disclaimer: evidence gathered from the forums of TheOneRing.net.