I think something like redbarret showed in the picture above would work with a modification. Instead of terminating at both pulleys as was suggested, simply loop around the second pulley, come back to the first, loop around the opposite way, and terminate it through another hole (or notch) on the other side.
For this to work, you would have to know how many revolutions the pulley makes to allow for the full distance of the y travel, which would be how many total loops to make (just as Claghorn suggested above). In addition, there would have to be enough width on the pulley to accommodate all of the loops without overlapping. Just by guessing at the size of the pulley, I would say no more than 8 loops would be needed, possibly less.
The same thing could be accomplished using a closed loop with the knot at the center of the loops on the pulley. If aligned/positioned properly with respect to the mid point of y travel, the knot would never leave the pulley. Therefore, it would not affect movement.
SD Press
Modeling software: 123D Design