Exoplanet orbital eccentricities encode key information about planetary system formation and evolution. For M dwarf planets in particular, the proximity of the habitable zone to the host star makes eccentricity additionally important for understanding habitability. However, constraints on orbital eccentricity typically require long and resource-intensive radial velocity campaigns. We summarize an investigation to extract eccentricity measurements of 143 planets orbiting early- to mid-M dwarfs across 81 systems, employing only their light curves from Kepler and host star density constraints from spectroscopy and Gaia. The "photoeccentric effect," described in exoplanet literature, enables the measurement of orbital eccentricity from a careful combination of transit duration and Kepler's 3rd law: both are linked to the stellar density. A disagreement between the stellar density inferred from the measured transit duration, and the density from Kepler's 3rd law, points to a non-zero orbital eccentricity. Employing our machinery in a standardized way on each of the 143 exoplanets, we derive an eccentricity posterior distribution for each. Through a hierarchical Bayesian analysis, we apply these individual posteriors to derive a population-level eccentricity distribution for planets around early- to mid-M dwarfs. Comparisons between the eccentricity distribution for singly- and multiply-transiting systems, as well as the distribution between M dwarfs and FGK dwarfs, help illuminate the dynamical states of M dwarf planets at a population level.