Statewide vital records are available generally beginning after 16 August 1909, and they can be obtained from Vital Records; PO Box 570; Jefferson City, MO 65102.
At the city level, deaths were recorded in St. Louis from 1850 - 1910, and births from 12 July 1870 - 1910. Although the recording was quite sporadic. These records can be obtained from St. Louis City Vital Records; PO Box 14702; St. Louis, MO 63178. Marriage records from St. Louis County are kept from 1876 to present day. These are in an archival card file, with entries for brides and grooms.
In Kansas City, the birth and death records date from 1874, but recording at this time was sparse. These records can be obtained from Kansas City Vital Records, City Hall, 414 E. 12th Street, Kansas City, MO 64106.
In 1863, a non-compulsory birth registration law was put into effect. It stated that county recorders could upon request record births in the regular deed books, sometimes they were recorded in the marriage books. These records are not indexed and there was very little compliance.
A state law was passed in 1883 requiring the registration of all births and deaths at a county level. In 1893, this law was inadvertently repealed, and never reenacted. Some counties had high rates of compliance, while other records are no where to be found. The records that remain have been microfilmed and are available at the Missouri State Archives, Jefferson City, MO 65102
Marriage records are generally available from the date of county inception, although there was no law that required licensing until 26 June 1881. Prior to this date, the marriage could be recorded at any convenient courthouse, there was no requirement to be married in the county of residence. Most of these records are kept by the Recorder of Deeds, and are consistently complete and indexed. These records have also been filmed by the Missouri State Archives. 
Many books have been published with compiled vital records for various counties in Missouri. The Mid-Continent Public Library in Independence, MO, has many of these books in the genealogy reference collection. Some of them are available for inter-library loan in the AFRA Genealogy circulating collection. A catalog lists the titles available called "Genealogy from the Heartland". 
State-wide references:
Guide to Public Vital Statistic Records in Missouri, Missouri Historical Records Survey, WPA, (St. Louis, MO, 1941) lists Birth, Death, Marriage and Divorce records available by county and at the state level. 
Robert E. Parkin's Guide to Tracing Your Family Tree in Missouri, Robert E. Parkin, Genealogical Research & Productions, (St. Louis, MO, 1979), is an excellent reference work. It lists the records available at the county level. It also gives the titles of compiled books by county for birth, death, and marriage records. 
A Guide to County Records on Microfilm, Missouri State Archives, (Jefferson City, MO, 1998