Found a listing for the Nerf Super Soaker DartFire on Amazon. Looks like the orange area has a nozzle for water powered by pump-action blasting while the darts launch from their holders, though it is unclear what mechanism powers the darts to launch. Part of me suspects that the large purple trigger does double-duty to prime and launch darts with each pull, but am not 100% sure.
Update: if I read part of the description sooner, I would have noticed “No priming; just pull the trigger to keep firing darts.” Well, that answers that – trigger does all the work for launching the darts, but that also suggests that these darts won’t fire with much force, either.