We present first results of a four-epoch VLBA survey for nonthermal emission toward all 556 compact radio sources previously identified in a deep VLA survey of the Orion Nebula Cluster (ONC). We identify VLBA counterparts toward an unprecedented 123 sources. Of these, 41 do not have X-ray counterparts, of which 34 also do not display near-infrared counterparts. Since these cannot be explained by extragalactic background sources, this suggests a component of the ONC population of young stellar objects that are too deeply embedded for even X-rays to be detectable. We find pervasive variability and detect even most of the highest-S/N sources in only one out of four epochs. Neither a negative spectral index nor extreme variability in the VLA data is a good predictor of a VLBA detection.