Learn Russian: Инвентаризация закончится через неделю.

Because закончить is a transitive verb, meaning it needs a direct object (someone finishes a specific task). Here, the stocktaking isn't finishing something else; it is coming to an end itself. We use the reflexive suffix -ся to make it intransitive, which is very common for events that conclude on their own: инвентаризация закончится (stocktaking will end).

I know через usually means across or through. How does it mean 'in' here?

When talking about time, через followed by the Accusative case indicates when a future event will happen, meaning 'in an amount of time from now'. So, через неделю means 'in a week'. It is important not to confuse this with за неделю, which would mean the actual process of stocktaking takes a week of continuous effort to complete.

Why is the perfective verb закончится used instead of the imperfective заканчивается?

The perfective aspect focuses on the completion or the result of an action. The sentence states a one-time fact about when the stocktaking will reach its final conclusion in the future, rather than describing the ongoing, repetitive process of it winding down, so the perfective закончиться is required.