We demonstrate a new method for measuring the escape fraction of ionizing photons using Hubble Space Telescope imaging of resolved stars in NGC 4214, a local analog of high-redshift starburst galaxies that are thought to be responsible for cosmic reionization. Specifically, we forward model the UV through near-IR spectral energy distributions of ∼83,000 resolved stars to infer their individual ionizing flux outputs. We constrain the local escape fraction by comparing the number of ionizing photons produced by stars to the number that are either absorbed by dust or consumed by ionizing the surrounding neutral hydrogen in individual star-forming regions. We find substantial spatial variation in the escape fraction (0%-40%). Integrating over the entire galaxy yields a global escape fraction of 25-15+16%. This value is much higher than previous escape fractions of zero reported for this galaxy. We discuss the sources of this apparent tension and demonstrate that the viewing angle and 3D interstellar medium geometric effects are the cause. If we assume that NGC 4214 has no internal dust, like many high-redshift galaxies, we find an escape fraction of 59% (an upper limit for NGC 4214). This is the first nonzero escape fraction measurement for UV-faint (MFUV =-15.9) galaxies at any redshift and supports the idea that starburst UV-faint dwarf galaxies can provide a sufficient number of ionizing photons to the intergalactic medium.