Molecular crystals such as polyethylene are of intense interest as flexible thermal conductors, yet their intrinsic upper limits of thermal conductivity remain unknown. Here, we report a study of the vibrational properties and lattice thermal conductivity of a polyethylene molecular crystal using an ab initio approach that rigorously incorporates nuclear quantum motion and finite temperature effects. We obtain a thermal conductivity along the chain direction of around 160 W m^{-1} K^{-1} at room temperature, providing a firm upper bound for the thermal conductivity of this molecular crystal. Furthermore, we show that the inclusion of quantum nuclear effects significantly impacts the thermal conductivity by altering the phase space for three-phonon scattering. Our computational approach paves the way for ab initio studies and computational material discovery of molecular solids free of any adjustable parameters.