I may have just found out who had the first car in Monroe! It may have happened all the way back in 1901! This little society article appeared in the Times-Picayune (New Orleans) December 9, 1900, page 15:
Miss Laura Wright, one of our most popular and beautiful young women, has won an automobile in Pueblo, Colo., where she visited her sister the past summer. It is worth about $1000, and was offered by the merchants of that city. Over 125,000 tickets were out. Miss Wright has ordered it shipped to her, and will have the distinction of introducing the horseless carriage in Monroe.
Way to go Laura!