"Leia gets shot down while making a run for it in the opening of A New Hope (it was only a stunner shot but it probably still gave her a headache) and then in Return of the Jedi Leia gets shot in the arm during the Endor battle. Although Han, Luke, Chewbacca, and even Lando take some abuse, they never get shot by the Storm Troopers. "

It's all about payback. They are gunning for her because, as you all should well know, she's the only main character in the SW films that doesn't miss what she's aiming at.