If you use our civilisation as a guide, in earlier years, then it becomes apparent that plastics without petrochemicals, are indeed possible. This becomes important if you are designing a society that is not industrialised, or does not use a petrochemical based technology (and never has), yet, has advanced to the stage of using plastics, and plastic-like materials.
In the 1800s, Mary Coad, of London, England, played about with a formulae for making a new ceramic. Coadstone had the plastisity of plastic, and could be shaped and moulded into the tiniest, spindilest shapes, before drying and becoming hard, yet still feeling a plasticy texture. Able to create far finer detail than concrete or marble ever could, almost every architectual feature in London from the late 1800s is coadstone.
That this ceramic material was possible to maufacture with simple hand tools, is known. The actual recipe is not. Mary Coad took it with her to the grave. Whilst there has been experimentation ever since, no-one else has ever found the formulae to recreate coadstone, only lesser imitants.
The proof we have of the material, lays in the creations she made, and the strange plastic-like properties they demonstrate.
Such a material existing in a ceramic and textile-based society is thus very possible indeed.
It should be noted that very little exists in the way of information on Coadstone, beyond hundreds of sites of historic buildings showcasing their pieces. The author herself, only first stumbled across it from a TV broadcast of "antiques challenge" and a soundbite, her research stemming from there.
Out with the new, in with the old (part-way down)