We perform a systematic Bayesian analysis of rotation vs. dispersion support (v_(rot)/σ) in 40 dwarf galaxies throughout the Local Volume (LV) over a stellar mass range 10^(3.5) M_⊙ < M⋆ < 10^8 M_⊙. We find that the stars in ∼90% of the LV dwarf galaxies studied -- both satellites and isolated systems -- are dispersion-supported. In particular, we show that 7/10 *isolated* dwarfs in our sample have stellar populations with v_(rot)/σ<0.6. All have v_(rot)/σ≲2. These results challenge the traditional view that the stars in gas-rich dwarf irregulars (dIrrs) are distributed in cold, rotationally-supported stellar disks, while gas-poor dwarf spheroidals (dSphs) are kinematically distinct in having dispersion-supported stars. We see no clear trend between v_(rot)/σ and distance to the closest L⋆ galaxy, nor between v_(rot)/σ and M⋆ within our mass range. We apply the same Bayesian analysis to four FIRE hydrodynamic zoom-in simulations of isolated dwarf galaxies (10^9M⊙<M_(vir)<10^(10)M⊙) and show that the simulated *isolated* dIrr galaxies have stellar ellipticities and stellar v_(rot)/σ ratios that are consistent with the observed population of dIrrs *and* dSphs without the need to subject these dwarfs to any external perturbations or tidal forces. We posit that most dwarf galaxies form as puffy, dispersion-supported systems, rather than cold, angular momentum-supported disks. If this is the case, then transforming a dIrr into a dSph may require little more than removing its gas
