Conway is a city in Laclede County, Missouri, United States. The population was 788 at the 2010 census. Conway is located 15 miles …Conway is a city in Laclede County, Missouri, United States. The population was 788 at the 2010 census. Conway is located 15 miles southwest of Lebanon. Conway was platted in 1869, and named after a local merchant. A post office has been in operation at Conway since 1871.