I posted an ad on craigslist in the free section, asking for old lumber. I got a bunch of old fence pickets and other lumber and built a nice coop and didn't spend a penny on wood.
Yes, and some communities also have "Freecycle" groups whose members offer, receive, and swap all kinds of things. I lined the interior of my coop with used vinyl vented trailer skirting panels. They're durable, waterproof, washable, ventilated — and free.
The overarching principle, of course, is allowing enough time to obtain free materials, and using your imagination for ways you can use them.
But if people are in a big hurry, it isn't possible and they end up spending a lot of money. That's why the "cost" of building a coop varies so widely.
And hey, frugality is fun!