We present the first results from the Quasar Feedback Survey, a sample of 42 z < 0.2, [O III] luminous AGNs (L-[O III] > 10(42.1) ergs s(-1)) with moderate radio luminosities (i.e. L-1.4GHz > 10(23.4) W Hz(-1); median L-1.4GHz = 5.9 x 10(23) W Hz(-1)). Using high spatial resolution (similar to 0.3-1 arcsec), 1.5-6 GHz radio images from the Very Large Array, we find that 67 percent of the sample have spatially extended radio features on similar to 1-60 kpc scales. The radio sizes and morphologies suggest that these may be lower radio luminosity versions of compact, radio-loud AGNs. By combining the radio-to-infrared excess parameter, spectral index, radio morphology, and brightness temperature, we find radio emission in at least 57 percent of the sample that is associated with AGN-related processes (e.g. jets, quasar-driven winds, or coronal emission). This is despite only 9.5-21 percent being classified as radio-loud using traditional criteria. The origin of the radio emission in the remainder of the sample is unclear. We find that both the established anticorrelation between radio size and the width of the [O III] line, and the known trend for the most [O III] luminous AGNs to be associated with spatially extended radio emission, also hold for our sample of moderate radio luminosity quasars. These observations add to the growing evidence of a connection between the radio emission and ionized gas in quasar host galaxies. This work lays the foundation for deeper investigations into the drivers and impact of feedback in this unique sample.