We present a numerically exact study of charge transport and its fluctuations through a molecular junction driven out of equilibrium by a bias voltage, using the inchworm quantum Monte Carlo method. After showing how the technique can be used to address any lead geometry, we concentrate on one dimensional chains as an example. The finite bandwidth of the leads is shown to affect transport properties in ways that cannot be fully captured by quantum master equations: in particular, we reveal an interaction-induced broadening of transport channels that is visible at all voltages and show how fluctuations of the current are a more sensitive probe of this effect than the mean current.