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!
November 15, 2021 edit and update:
To get an idea of the look of cars in 1900, look here: Early 1900s cars .
Research in 1900 – 1901 Pueblo, CO papers stated she had bought a diamond ring from The Churchill Jewelry Company and it must have given her an entry into the drawing. A February 2, 1901 article in the Indicator claimed the vehicle was still being held by relatives in Pueblo but would soon be shipped to Louisiana.