We have theoretically investigated the feasibility of constructing a spintronic field-effect transistor with the active channel made of a polymer chain with the antiferromagnetic coupling oriented in the source-to-drain direction. We found two different device function regimes controlling the on-chain spin-charge carrier density by tuning the gate voltage. At higher charge carrier densities, the source-drain current linearly increases with decreasing charge carrier densities. In this regime, no polymer spin-polarized current is observed. Upon reaching a critical gate voltage, the current decreases with decreasing charge densities. It is accompanied by the formation of spin-polarized current, generated by an on-chain process, which can be related to spin-charge spatial distribution symmetry breaking caused either by an application of the source-to-drain voltage (higher spin polarization near the drain), or the breakdown of the Peierls dimerization near chain ends. Numerical simulation of the transistor characteristics suggests that the design of a polymer spintronic field-effect transistor is in principle feasible.