The structure of a dwarf galaxy is an important probe of the effects of stellar feedback and environment. Using an unprecedented sample of 223 low-mass satellites from the ongoing Exploration of Local Volume Satellites survey, we explore the structures of dwarf satellites in the mass range 10^5.5^<M_*_<10^8.5^M_{sun}_. We survey satellites around 80% of the massive, M_K_{<}-22.4mag, hosts in the Local Volume (LV). Our sample of dwarf satellites is complete to luminosities of M_V_{<}-9mag and surface brightness {mu}_0,V_<26.5mag/arcsec^2^ within at least ~200 projected kpc of the hosts. For this sample, we find a median satellite luminosity of M_V_=-12.4mag, median size of r_e_=560pc, median ellipticity of {epsilon}=0.30, and median Sersic index of n=0.72. We separate the satellites into late- and early-type (29.6% and 70.4%, respectively). The mass-size relations are very similar between them within ~5%, which indicates that the quenching and transformation of a late-type dwarf into an early-type one involves only very mild size evolution. Considering the distribution of apparent ellipticities, we infer the intrinsic shapes of the early- and late-type samples. Combining with literature samples, we find that both types of dwarfs are described roughly as oblate spheroids that get more spherical at fainter luminosities, but early-types are always rounder at fixed luminosity. Finally, we compare the LV satellites with dwarf samples from the cores of the Virgo and Fornax clusters. We find that the cluster satellites show similar scaling relations to the LV early-type dwarfs but are roughly 10% larger at fixed mass, which we interpret as being due to tidal heating in the cluster environments. The dwarf structure results presented here are a useful reference for simulations of dwarf galaxy formation and the transformation of dwarf irregulars into spheroidals.