I have a Staff of the Woodlands, which allows me to cast the spell Awaken. Awaken has an expensive material component of "an agate worth at least 1,000 gp, which the spell consumes". Is that agate required and consumed when Awaken is cast through the Staff, or does casting it through the Staff somehow modify or remove that requirement?
No they do not require components at all including somatic or verbal for that matter. So casting an Arcane Lock from a Staff of the Magi doesn't require the 25g worth of gold dust nor would the Staff of the Woodlands require it for Awaken. It may seem "overpowered" but the typical druid doesn't go around making sentient armies of animals for good reason.
DMG p141 the section on magical items and casting spells from them.
The spell is cast at the lowest possible spell level, doesn't expend any of the user's spell slots, and requires no components, unless the item's description says otherwise.