Wicker would be my first choice. Lightweight and compostable! It is hard to find one that's big
enough and sturdy enough. I'm sure it's out there.
Laundry bags are an option, are these sold as laundry bags? I'd prefer to order something like a cloth bag over the internet even though wicker seems best because overpackaging is a pet peeve of mine and I'm imagining a really massive box for a laundry-sized basket.
We get our laundry pretty dirty here! Jeans age well, but the baby's cloth diapers are another story...