First I'd ream out the rear sprayer nozzle with a straight pin. The most inconsequential amount of road muck can plug that nozzle up.

If that doesn't work, then it comes down to partially removing the headliner to access the supply hose from underneath. Not a fun job by any means.

OTOH, if you're in the snow belt, this could just as easily be ice in the lines (unless the garage for your car is heated).