We present ALMA observations of the Orion Nebula that cover the OMC1 outflow region. Our focus in this paper is on compact emission from protoplanetary disks. We mosaicked a field containing ~600 near-IR-identified young stars, around which we can search for sub-millimeter emission tracing dusty disks. Approximately 100 sources are known proplyds identified with the Hubble Space Telescope. We detect continuum emission at 1mm wavelengths toward ~20% of the proplyd sample, and ~8% of the larger sample of near-IR objects. The noise in our maps allows 4{sigma} detection of objects brighter than ~1.5mJy, corresponding to protoplanetary disk masses larger than 1.5M_J_ (using standard assumptions about dust opacities and gas-to-dust ratios). None of these disks are detected in contemporaneous CO(2-1) or C^18^O(2-1) observations, suggesting that the gas-to-dust ratios may be substantially smaller than the canonical value of 100. Furthermore, since dust grains may already be sequestered in large bodies in Orion Nebula cluster (ONC) disks, the inferred masses of disk solids may be underestimated. Our results suggest that the distribution of disk masses in this region is compatible with the detection rate of massive planets around M dwarfs, which are the dominant stellar constituent in the ONC.