I honestly don't think there is anything that would take the place of leather. Anything poly, plastic, foam, are all petroleum based so that doesn't work either.
perhaps you can make some of that home made clay-dough with flour, water, salt and something else. But you would want to make sure that it is very well sealed once it's baked.
Plaster might work if you carve that then saturate a pieces of fabric with starch then overly the carved plaster with the cloth and sort of wet form it into the shapes. Again this is going to be stiff... but you might be able to stuff the back sides with cotton batting and use some sort of natural woven cloth as a stabilizer to keep it in place.
I originally thought layers of Felt would work but true felt is (you guessed it) 'animal based' and the synthetic felts are uh huh... petroleum based. If your customer is not opposed to non-kill animal products like wool... that might work. Wool felt once dampened can be compressed and it will retain the shape when it's dry.