Van der Waals heterostructures based on graphene and inorganic two-dimensional (2D) crystals enable exquisite control of interlayer coupling, and emergent electronic and optical phenomena^1,2. Extending this concept to organic 2D crystals has been hindered by weak, non-directional interlayer interactions that frustrate lattice registry. Here we introduce a bottom-up strategy for programmable lattice engineering in organic van der Waals heterostructures of 2D polymers. Sequential on-water-surface assembly enables layer-by-layer stacking of chemically distinct 2D polymers with defined lattice registry, stacking sequence and thickness, yielding both lattice-matched and controlled lattice-mismatched heterostructures. Structural characterization reveals commensurate epitaxy in lattice-matched and small-mismatched systems, whereas large-mismatch interfaces exhibit moiré features and strain-relief distortions. Ultrafast spectroscopy demonstrates efficient interfacial charge separation and first-principles calculations reveal built-in electric fields and interfacial potential steps arising from interfacial dipole alignment. Devices exhibit diode-like rectification ratios exceeding 10⁷ that systematically decrease with increasing lattice mismatch, establishing organic 2D polymer van der Waals heterostructures as a lattice-engineered platform for (opto)electronic and quantum phenomena.