Mode-locked vertical external-cavity surface emitting lasers are promising compact sources for high-power, ultrafast pulses with excellent beam quality and the flexibility offered by an external cavity. Classical models of these lasers use either phenomenological approaches, which rely heavily on experimentally observed macroscopic parameters, or are based on quasi-equilibrium conditions. Although these models enjoy widespread success, they cannot capture the underlying charge carrier dynamics, shown to be critical components of pulse formation and propagation. The Maxwell Semiconductor Bloch Equations capture these dynamics through a coupling of pulse propagation to the field induced polarization within an active semiconductor quantum well. We utilize a transverse implementation of this model to microscopically investigate fundamental Gaussian pulse formation as well as destabilizing effects of pump parameters. These behaviors are directly linked to the underlying charge carrier dynamics. Excess carriers around the pulse's spatial or spectral centers destabilizes the pulse and are shown to lead to the formation of higher order transverse modes and secondary pulses within the cavity.