We present a morphological study of the 17 lensed Ly alpha emitter (LAE) galaxies of the Baryon Oscillation Spectroscopic Survey Emission-Line Lens Survey (BELLS) for the GALaxy-Ly alpha EmitteR sYstems (BELLS GALLERY) sample. This analysis combines the magnification effect of strong galaxy-galaxy lensing with the high resolution of the Hubble Space Telescope to achieve a physical resolution of similar to 80 pc for this 2 < z < 3 LAE sample, allowing a detailed characterization of the LAE rest-frame ultraviolet continuum surface brightness profiles and substructure. We use lens-model reconstructions of the LAEs to identify and model individual clumps, which we subsequently use to constrain the parameters of a generative statistical model of the LAE population. Since the BELLS GALLERY sample is selected primarily on the basis of Ly alpha emission, the LAEs that we study here are likely to be directly comparable to those selected in wide-field, narrowband LAE surveys, in contrast with the lensed LAEs identified in cluster-lensing fields. We find an LAE clumpiness fraction of approximately 88%, which is significantly higher than that found in previous (non-lensing) studies. We find a well-resolved characteristic clump half-light radii of similar to 350 pc, a scale comparable to the largest H II regions seen in the local universe. This statistical characterization of LAE surface-brightness profiles will be incorporated into future lensing analyses using the BELLS GALLERY sample to constrain the incidence of dark-matter substructure in the foreground lensing galaxies.