  The structure of a dwarf galaxy is an important probe into the effects of
stellar feedback and environment. Using an unprecedented sample of 223 low-mass
satellites from the ongoing Exploration of Local VolumE Satellites (ELVES)
Survey, we explore the structures of dwarf satellites in the mass range
$10^{5.5}<M_\star<10^{8.5}$M$_\odot$. We survey satellites around $80\%$ of the
massive, $M_K<-22.4$ mag, hosts in the Local Volume. Our sample of dwarf
satellites is complete to luminosities of $M_V<-9$ mag and surface brightness
$\mu_{0,V}<26.5$ mag arcsec$^{-2}$ within at least $\sim200$ projected kpc. We
separate the satellites into late- and early-type, finding the mass-size
relations are very similar between them, to within $\sim5\%$. This similarity
indicates that the quenching and transformation of a late-type dwarf into an
early-type 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 get thicker 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.
