Pretty much all chemical changes can be made with reasonable amounts of energy. That includes making "bioplastics" as well as the typical plastics we use today like polyethylene, polystyrol and so on, from biomaterials. What doesn't work in a way that's remotely economical is transmuting elements. It is, for example, possible to make gold today, the old dream of alchemists. But it's several orders of magnitude too expensive.

Common plastics are made from highly abundant elements, so running out of oil as a chemical feedstock is a quite surmountable problem given cheap enough energy.