The M-W also offers the definition you quoted. Let me rephrase my question. Is "coop" functioning as a noun or verb? In my opinion, I don't think it's a verb in the given context. But I don't understand how it can work as a noun either. Help me, please.
home coop is a compound noun (two nouns) meaning the coop where they have their home - their coop of origin, so to speak, or the coop that serves as their "headquarters". There are many compounds with home where home has the same meaning.
Compare:
home country, home state, home city, home town, home port, and even, in sc