So, how do we produce polymers from water and carbon dioxide? By turning them into synthesis gas, a mixture of carbon monoxide and hydrogen.

Given a supply of water (and of course energy) one can use processes similar to those used to produce return-voyage propellants in the Mars Direct architecture -- namely, the reverse water gas shift. The water is split via electrolysis to obtain hydrogen, some of which is then reacted with atmospheric carbon dioxide via RWGS to produce carbon monoxide and water. The water is returned to the cycle and the freed oxygen can be used for life support, but the carbon monoxide is mixed with the remaining hydrogen to produce synthesis gas, which then feeds into the production of polymer materials (via free radical polymerization, Ziegler-Natta polymerization, metallocene catalysis polymerization, etc.).

The beauty of martian plastics is that they can be produced in part with equipment which is already present on the surface for other purposes, using processes which are well-known from decades of terrestrial industrial experience.