We perform a systematic Bayesian analysis of rotation vs. dispersion support ($v_{rm rot} / sigma$) in $40$ dwarf galaxies throughout the Local Volume (LV) over a stellar mass range $10^{3.5} M_{rm odot} < M_{star} < 10^8 M_{rm odot}$. We find that the stars in $sim 80%$ of the LV dwarf galaxies studied -- both satellites and isolated systems -- are dispersion-supported. In particular, we show that $6/10$ *isolated* dwarfs in our sample have $v_{rm rot} / sigma < 1.0$. All have $v_{rm rot} / sigma lesssim 2.0$. 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 $v_{rm rot} / sigma$ and distance to the closest $rm L_{star}$ galaxy, nor between $v_{rm rot} / sigma$ and $M_{star}$ within our mass range. We apply the same Bayesian analysis to four FIRE hydrodynamic zoom-in simulations of isolated dwarf galaxies ($10^9 M_{odot} < M_{rm vir} < 10^{10} M_{rm odot}$) and show that the simulated *isolated* dIrr galaxies have stellar ellipticities and stellar $v_{rm rot} / sigma$ 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-dominated 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.
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