By definition, super-conductor is a conductor no matter what materials it is made from.
So, IF someone could **really turn a plastic (e.g. those ferro-plastic w/ free electrons flowing around) into superconductor, the materials (if that condition persists) will sure conduct electricity.
Though, in reality, the structure in plastic is too complicated (generally chains of hydrocarbons) that will prohibit entering the super-conductivity state.