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We present the first results of a multi-year program to map the orbits of M dwarf multiples within 25 parsecs. The observations were conducted primarily during 2019 - 2020 using speckle interferometry at the Southern Astrophysical Research (SOAR) Telescope in Chile, using the High-Resolution Camera mounted on the adaptive optics module (HRCam+SAM). The sample of nearby M dwarfs is drawn from three sources: multiples from the RECONS long-term astrometric monitoring program at the SMARTS 0.9m, known multiples for which these new observations will enable or improve orbit fits, and candidate multiples flagged by their astrometric fits in Gaia Data Release 2 (DR2). We surveyed 333 of our 338 M dwarfs via 830 speckle observations, detecting companions for 63% of the stars. Most notably, this includes new companions for 76% in the subset selected from Gaia DR2. In all, we report the first direct detections of 97 new stellar companions to the observed M dwarfs. Here we present the properties of those detections, the limits of each non-detection, and five orbits with periods 0.67 - 29 yr already observed as part of this program. Companions detected have projected separations of 0.024 - 2.0 arcsec (0.25 - 66 AU) from their primaries and have $ΔI \lesssim 5.0$ mag. This multi-year campaign will ultimately map complete orbits for nearby M dwarfs with periods up to 3 yr, and provide key epochs to stretch orbital determinations for binaries to 30 yr.