If you are looking for Missouri background check resources, you may have a few different options to pursue.
Family records such as marriage or divorce documents are not given out to the general public, and you can only get copies if you are a family member (and proof is required). Requests for these can go to the local county clerk or the Bureau of Vital Records in Jefferson City.
Getting a Criminal Check
A more helpful Missouri background check resource is the Criminal Justice system. The have an online system called the Missouri Automated Criminal History Site (MACHS) that allows for name-based searches of their database. Each search will cost you $10 and you pay online via credit card. You can find convictions, and very recent arrests. If you have the individual's permission and fingerprints, more information will be available to you.