We present a sample of nearby dwarf galaxies with radio-selected accreting massive black holes (BHs), the majority of which are non-nuclear. We observed 111 galaxies using sensitive, high-resolution observations from the Karl G. Jansky Very Large Array (VLA) in its most extended A-configuration at X band (~8-12GHz), yielding a typical angular resolution of ~0.25" and rms noise of ~15{mu}Jy. Our targets were selected by crossmatching galaxies with stellar masses M_*_<=3x10^9^M_{sun}_ and redshifts z<0.055 in the NASA-Sloan Atlas with the VLA Faint Images of the Radio Sky at Twenty centimeters Survey. With our new high-resolution VLA observations, we detect compact radio sources toward 39 galaxies and carefully evaluate possible origins for the radio emission, including thermal HII regions, supernova remnants, younger radio supernovae, background interlopers, and active galactic nuclei (AGNs) in the target galaxies. We find that 13 dwarf galaxies almost certainly host active massive BHs, despite the fact that only one object was previously identified as having optical signatures of an AGN. We also identify a candidate dual radio AGN in a more massive galaxy system. The majority of the radio-detected BHs are offset from the center of the host galaxies, with some systems showing signs of interactions/mergers. Our results indicate that massive BHs need not always live in the nuclei of dwarf galaxies, confirming predictions from simulations. Moreover, searches attempting to constrain BH seed formation using observations of dwarf galaxies need to account for such a population of "wandering" BHs.