337th Infantry was organic to 85th Infantry Division. Company L was an element of the regiment's 3d Battalion, one of its three rifle companies. The company was serving in Italy during the period you've indicated. There are a number of division histories available but most too general to enable you to follow a company's activities/movements with any accuracy. Perhaps the regimental veterans association can assist.
337th Infantry Reunion Association c/o William C. Mitchell 2332 21st Street, Southwest Akron, OH 44314-1921 (330)753-6045