The Delaware Basin, Texas is currently a hot-spot of induced seismicity and ground deformation due to fluid extraction and injection associated with horizontal drilling techniques; however, the driving mechanism behind the seismicity and deformation remains under debate. Using vertical and east-west horizontal surface deformation measurements derived from Sentinel-1 InSAR, we show that the subsurface responds differently to oil and gas activity in the northern and southeastern portions of the basin. In the north, where there is little seismicity, deformation patterns display long-wavelengths and equidimensional patterns. In contrast, the southeast region hosts most of the seismicity and displays spatial deformation patterns with narrow linear features that strike parallel to the maximum principal horizontal stress and to trends in seismicity, suggesting movement along normal faults. We model a linear deformation feature using edge dislocations and show that the InSAR observations can be reproduced by slip on normal faults contained within the Delaware Mountain Group (DMG), the formation that hosts local wastewater injection and the majority of earthquakes. Our model consists of three parallel, high-angle normal faults, with two dipping toward one another in a graben structure. Slip magnitudes reach up to 27.5 cm and are spatially correlated with injection wells. Measured seismicity can only explain ~2% of the fault motion predicted by our fault model, suggesting that slip leading to the deformation is predominantly aseismic. We conclude that seismic and aseismic fault motion in the southeastern Delaware Basin is likely driven by wastewater injection near critically-stressed normal faults within the DMG.