We investigate the potential of photonic lantern (PL) fiber-fed spectrometers for two-dimensional spectroastrometry. Spectroastrometry, a technique for studying small angular scales by measuring centroid shifts as a function of wavelength, is typically conducted using long-slit spectrographs. However, slit-based spectroastrometry requires observations with multiple position angles to measure two-dimensional spectroastrometric signals. In a typical configuration of PL-fed spectrometers, light from the focal plane is coupled into the few-moded PL, which is then split into several single-mode outputs, with the relative intensities containing astrometric information. The single-moded beams can be fed into a high-resolution spectrometer to measure wavelength-dependent centroid shifts. We perform numerical simulations of a standard six-port PL and demonstrate its capability of measuring spectroastrometric signals. The effects of photon noise, wavefront errors, and chromaticity are investigated. When the PL is designed to have large linear responses to tip tilts at the wavelengths of interest, the centroid shifts can be efficiently measured. Furthermore, we provide mock observations of detecting accreting protoplanets. PL spectroastrometry is potentially a simple and efficient technique for detecting spectroastrometric signals.