  We present results from a search for radio emission in 77 stellar systems
hosting 140 exoplanets, predominantly within 17.5 pc using the Very Large Array
(VLA) at $4-8$ GHz. This is the largest and most sensitive search to date for
radio emission in exoplanetary systems in the GHz frequency range. We obtained
new observations of 58 systems, and analyzed archival observations of an
additional 19 systems. Our choice of frequency and volume limit are motivated
by radio detections of ultracool dwarfs (UCDs), including T dwarfs with masses
at the exoplanet threshold of $\sim\!13\,M_J$. Our surveyed exoplanets span a
mass range of $\approx\,10^{-3}-10\,M_J$ and semi-major axes of
$\approx\,10^{-2}-10\,$AU. We detect a single target - GJ 3323 (M4) hosting two
exoplanets with minimum masses of 2 and 2.3$\,M_\oplus$ - with a circular
polarization fraction of $\approx\,40\%$; the radio luminosity agrees with its
known X-ray luminosity and the G\"udel-Benz relation for stellar activity
suggesting a likely stellar origin, but the high circular polarization fraction
may also be indicative of star-planet interaction. For the remaining sources
our $3\sigma$ upper limits are generally
$L_\nu\lesssim\,10^{12.5}\,\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{Hz}^{-1}$,
comparable to the lowest radio luminosities in UCDs. Our results are consistent
with previous targeted searches of individual systems at GHz frequencies while
greatly expanding the sample size. Our sensitivity is comparable to predicted
fluxes for some systems considered candidates for detectable star-planet
interaction. Observations with future instruments such as the Square Kilometer
Array and Next Generation Very Large Array will be necessary to further
constrain emission mechanisms from exoplanet systems at GHz frequencies.
