Emission line galaxies provide a crucial tool for the study of galaxy formation and evolution, providing a means to trace a galaxy's star formation history or metal enrichment, and to identify galaxies at a range of stellar masses. In this paper, we present a study of emission line galaxies in the Survey for IIigh-z Absorption Red and Dead Sources (SHARDS) Frontier Fields (FF) medium-band survey. Through detailed flux calibrations we combine the first results of the SHARDS-FF survey with existing Hubble Frontier Field data to select 1098 candidate emission line galaxies from the Hubble Frontier Filed clusters Abell 370 and MACS J1149.5+2223. Furthermore, we implement this deep medium-band imaging to update photometric redshift estimates and stellar population parameters and discover 38 predominantly low-mass H alpha emitters at redshifts 0.24 < z < 0.46. Overall, 27 of these sources have corresponding ultraviolet (UV) data from the Hubble Space Telescope that allow us to distinguish these sources and investigate the burstiness of their star formation histories. We find that more than 50 per cent of our sample shows an enhancement in H alpha over UV, suggesting recent bursts in star formation on time-scales of a few to tens of Myr. We investigate these sources and find that they are typically low-mass discy galaxies with normal sizes. Their structures and star formation suggest that they are not undergoing mergers but are bursting due to alternative causes, such as gas accretion.