Despite many attempts to detect radio emissions from magnetized exoplanets, a reproducible, unambiguous detection remains elusive. On the other hand, the search for periodic radio emissions from exoplanet host stars that may be induced by star-planet interactions in, for example, hot Jupiter systems, has only recently begun. In this third paper of the ROME (Radio Observations of Magnetized Exoplanets) series, we present the results of a targeted radio survey of 17 nearby systems that host exoplanet, brown dwarf, and low-mass stellar companions conducted with the Arecibo radio telescope at ∼5 GHz. This GHz-frequency survey has the greatest frequency coverage of any to date, while providing mJy-level sensitivity over <1 s integration times. Neither auroral radio emissions from the substellar targets, nor exoplanet-induced stellar radio bursts were detected. These results are considered within the context of observed brown dwarf radio emissions, which support magnetospheric phenomenon similar to that found at Jupiter. We also describe the orbital phase coverage of our data for systems that may feature star-planet interactions, and briefly examine our results within the context of other searches for star-planet interactions within the same systems. Finally, as several of our targets are within their respective systems' habitable zones, we consider the implications of our survey on the search for technosignatures within those systems.