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The Transiting Exoplanet Survey Satellite (TESS) is a NASA Explorer-class mission designed for finding exoplanets around nearby stars. TESS image data can also serve as a valuable resource for asteroid and comet detection, including near-Earth objects (NEOs). In order to exploit the TESS image data for moving object detection and potential object discovery, our team has developed an image processing pipeline as part of the Lincoln Near-Earth Asteroid Research (LINEAR) program, sponsored by the NASA NEO Observations Program. The LINEAR-TESS pipeline is currently in operation and reporting asteroid observations to the Minor Planet Center. In this paper we discuss the algorithms and methodology utilized to push the limits of the astrometric accuracy and photometric sensitivity of the TESS instrument for asteroid detection without a priori information on the ephemerides of the objects, and report on observation statistics from the first two years of TESS mission data.