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Spinspotter is a robust and automated algorithm designed to extract stellar rotation periods from large photometric datasets with minimal supervision. Our approach uses the autocorrelation function (ACF) to identify stellar rotation periods up to one-third the observational baseline of the data. Our algorithm also provides a suite of diagnostics that describe the features in the ACF, which allows the user to fine-tune the tolerance with which to accept a period detection. We apply it to approximately 130,000 main-sequence stars observed by the Transiting Exoplanet Survey Satellite (TESS) at 2-minute cadence during Sectors 1-26, and identify rotation periods for 13,504 stars ranging from 0.4 to 14 days. We demonstrate good agreement between our sample and known values from the literature and note key differences between our population of rotators and those previously identified in the Kepler field, most notably a large population of fast-rotating M dwarfs. Our sample of rotating stars provides a data set with coverage of nearly the entire sky that can be used as a basis for future gyrochronological studies, and, when combined with proper motions and distances from Gaia, to search for regions with high densities of young stars, thus identifying areas of recent star formation and undiscovered moving group members. Our algorithm is publicly available for download and use on GitHub.