We present the properties of the globular clusters (GCs) and nuclear star clusters (NSCs) of low-mass (10^5.5^<M_*_<10^8.5^M_{sun}_) early-type satellites of Milky Way-like and small group hosts in the Local Volume (LV) using deep, ground-based data from the ongoing Exploration of Local VolumE Satellites Survey. This sample of 177 dwarfs significantly increases the statistics for studying the star clusters of dwarfs in low-density environments, offering an important comparison to samples from nearby galaxy clusters. The LV dwarfs exhibit significantly lower nucleation fractions at fixed galaxy mass than dwarfs in nearby clusters. The masses of NSCs of LV dwarfs show a similar scaling of M_*,NSC_{propto}M_*,gal_^0.4^ as that found in clusters but offset to lower NSC masses. To deal with foreground/background contamination in the GC analysis, we employ both a statistical subtraction and Bayesian approach to infer the average GC system properties from all dwarfs simultaneously. We find that the GC occupation fraction and average abundance are both increasing functions of galaxy stellar mass, and the LV dwarfs show significantly lower average GC abundance at fixed galaxy mass than a comparable sample of Virgo dwarfs analyzed in the same way, demonstrating that GC prevalence also shows an important secondary dependence on the dwarf's environment. This result strengthens the connection between GCs and NSCs in low-mass galaxies. We discuss these observations in the context of modern theories of GC and NSC formation, finding that the environmental dependencies can be well explained by these models.