The influence of local dynamics on measurements of $H_0$ has been an area of ongoing investigation. We present a novel Bayesian framework to account for non-linear peculiar velocity dynamics in the inference of $H_0$. We validate the framework on simulated type Ia supernovae, which have been emulated according to realistic host galaxy properties in an analogue of our Local Group. The framework enables the statistically principled treatment of observational errors and peculiar velocities in inferences of $H_0$ at the field level, i.e. exploiting information from the full three-dimensional velocity field. We find that unaccounted-for, non-linear velocity dynamics are unlikely to be the cause of the Hubble tension. The Bayesian model presented here paves the way for inferences of $H_0$ from supernovae at $z<0.023$.