We find that conjugated polymers can undergo reversible structural phase transitions during electrochemical oxidation and ion injection. We study poly[2,5-bis(thiophenyl)-1,4-bis(2-(2-(2-methoxyethoxy)ethoxy)ethoxy)benzene] (PB2T-TEG), a conjugated polymer with glycolated side chains. Using grazing incidence wide-angle X-ray scattering (GIWAXS), we show that, in contrast to previously known polymers, this polymer switches between two structurally distinct crystalline phases associated with electrochemical oxidation/reduction in an aqueous electrolyte. Importantly, we show that this unique phase change behavior has important physical consequences for ion-polaron pair transport. Notably, using moving front experiments visualized by both optical microscopy and super-resolution photoinduced force microscopy (PiFM), we show that a laterally propagating ion-polaron pair front in PB2T-TEG exhibits non-Fickian transport, retaining a sharp step-edge profile, in stark contrast to the Fickian diffusion more commonly observed in polymers like P3MEEMT. This structural phase transition is reminiscent of those accompanying ion uptake in inorganic materials like LiFePO4. We propose that the engineering of similar properties in future conjugated polymers may enable the realization of new materials with superior performance in electrochemical energy storage or neuromorphic memory applications.