We investigate the feasibility of using ballistic electron emission microscopy (BEEM) to study possible thickness variations of ultrathin SiO2, which might exist at substrate defects, such as steps. We find that simple BEEM imaging of the oxide film sandwiched into a metal-oxide-semiconductor (MOS) structure does not reveal any features that could be related to the oxide film. We further present initial results suggesting that hat-electron resonance in the oxide conduction band could be observed by BEEM and could be sensitive to local film thickness. We also confirm the ability of oxide film to sustain injection of very high densities of hot electrons without any observable damage. In some cases we observe local damage of the MOS structure induced by BEEM measurements, but we conclude that it is most likely related to failure of the metal overlayer and probably not related to hot-electron breakdown of the oxide.