I saw a bottle organ at Liverpool museum years ago, I wonder if that could be scaled down to accordion size, no problems with keeping in tune, unlike cane which is at the mercy of changes of air pressure, humidity,etc. The favourite cane for reedmakers is Arundo donax but anything that will vibrate and can be shaped can be used as reed material. Many woods lend themselves favourably to reed making. The Welsh Pibgorn uses a tube of Elder for its reed and a lot of modern make repro bagpipes now use 'plastic' reeds. Bamboo tends to be too hard and brittle but maybe could be used for free reeds although mounting it I think would be a problem.
Which instrument uses a paper reed? The Kazoo.