If the arrest was recent
Start with the county jail
Someone just arrested in Missouri is almost always held in a county jail run by the county sheriff, not in state prison.
County jail rosters are fragmented: some Missouri sheriffs post an online booking/inmate roster, many do not.
Missouri is a statewide VINE state — VINELink lets you check custody status and register for release notifications, and a large share of Missouri county jails plus the state prisons feed into it.
If a jail is not searchable online, call that county sheriff's jail booking or records line directly.
Many Missouri sheriffs publish their own jail roster (for example Greene County/Springfield, Jackson County, and St. Louis County), but coverage, format, and update frequency vary county to county; there is no single statewide county-jail roster.