This is a fun question. I am not against plastics in my designs but lets see what we can do to get around this!
Wood? I would think that a good hard wood with closely packed grain would be a great building block for grow beds and even the fish tank. If you employed basic boat building skills and adhesives such as a resin you might be "sustainable". You could use a trough design to move water from point A to point B...also achievable with wooden planks.
To get around the use of pumps you would probably need to employ something like a paddle wheels next to a flowing stream to carry water from a low point to a high point. From there I think gravity could be employed.
As far as some form of ebb and flow system you would need to get tricky with your siphon. Perhaps a loop siphon would be easier to achieve with natural pieces. Perhaps cured animal gut could be dried in a loop siphon form. But then again using bamboo to fashion a bell siphon might work pretty well.