You can find pulleys in old tape drives, and even some
computer disk drives use a belt and pulleys for the drawer mechanism. I
did a repower on a Riv U25C years ago with a small square belt and
pulleys that worked really well.
The
small pulley on the motor shaft and large one on the gearbox add some
extra reduction to the gearbox, but with a steamer all of your reduction
would be limited to the pulleys alone. That's probably going to make it
too fast, even if you make the pulley on the axle as close to the
driver diameter as possible. I would cut a groove for the belt into the
existing axle gear if I was to try it, though.