The tannic acid may come most often from oak galls, but may also be sourced from oak-apple, pistachio, or aleppo pine galls; bark from logwood/campeachy, pine, oak, chestnut, cherry, or mountain ash trees may be used; even such things as hemlock, pomegranate rinds, or horse chestnuts may be used to extract the tannin.
Here’s a video of the BBC’s introduction to the oak gall as it features in iron gall ink.