We present high spatial resolution maps of ro-vibrational molecular hydrogen emission from the environment of the GG Tau A binary component in the GG Tau quadruple system. The H2 v= 1-0 S(1) emission is spatially resolved and encompasses the inner binary, with emission detected at locations that should be dynamically cleared on several hundred-year timescales. Extensions of H2 gas emission are seen to ~100 AU distances from the central stars. The v = 2-1 S(1) emission at 2.24 microns is also detected at ~30 AU from the central stars, with a line ratio of 0.05 +/- 0.01 with respect to the v = 1-0 S(1) emission. Assuming gas in LTE, this ratio corresponds to an emission environment at ~1700 K. We estimate that this temperature is too high for quiescent gas heated by X-ray or UV emission from the central stars. Surprisingly, we find that the brightest region of H2 emission arises from a spatial location that is exactly coincident with a recently revealed dust "streamer" which seems to be transferring material from the outer circumbinary ring around GG Tau A into the inner region. As a result, we identify a new excitation mechanism for ro-vibrational H2 stimulation in the environment of young stars. The H2 in the GG Tau A system appears to be stimulated by mass accretion infall as material in the circumbinary ring accretes onto the system to replenish the inner circumstellar disks. We postulate that H2 stimulated by accretion infall could be present in other systems, particularly binaries and "transition disk" systems which have dust cleared gaps in their circumstellar environments.