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.
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