Her role is to oversee the town's power plant and although somewhat skilled, she is by no means a genius at repairing broken electronic or mechanical equipment and this can cause problems and results in her requesting you fix the power generator if Mira's deranged ex-husband, Jack, blows himself up. She is very fond of her robot companion, Toto, and they seemingly go everywhere together. It assists her in her daily chores around the plant.
Stop officer Jack from blowing up the power plant: Jack just went though a bad divorce and snapped, strapping himself with explosives and running into the power plant. Dorothy would be appreciative if the Chosen One made sure the power plant didn't blow up.
Her name, and her barking robot called Toto, are references to The Wizard of Oz, a 1939 film.
In one of her floats, Dorothy asks the player if they happen to have an illudium Q-38 space modulator; this references the cartoon series Marvin the Martian, where the titular character often attempts to destroy Earth using an "illudium Q-36 explosive space modulator".