Goniopolar metals exhibit opposite thermoelectric polarities along different crystallographic directions, enabling zero-field transverse thermoelectricity but offering few means for external control. Here we show that a spatially structured Laguerre–Gaussian vector potential can programmably reconstruct the goniopolar phase space of NaSn₂As₂. First-principles-derived Wannier transport with bond-dependent Peierls coupling reveals two OAM-dependent spatial scaling laws: the radial response follows the Laguerre–Gaussian radius rₘₐₓ∝√|ℓ|, while calculations for |ℓ|=2--5 yield a dominant angular harmonic m_dom=2|ℓ|, encoding the optical winding in a frequency-doubled thermoelectric response. Increasing |ℓ| simultaneously reconstructs pre-existing goniopolar windows, enhancing a representative window by approximately 17% at |ℓ|=5. By contrast, reversing ℓ at fixed polarization produces only a small correction that approximately interchanges upon polarization reversal. Energy-resolved transport reveals that the structured field redistributes in-plane and cross-plane electronic velocities, shifting the directional Seebeck-zero boundaries that define the goniopolar state. These results establish vortex position and OAM magnitude as programmable control coordinates for goniopolar thermoelectricity.