No. The Missouri River is a tributary of the Mississippi River—it flows into the Mississippi at St. Louis. A distributary is a branch that flows away from the main stem (e.g., in a delta). The Missouri has no distributaries; it is the largest tributary by length (2,341 mi) in the Mississippi system.
Downstream, the Mississippi does develop distributaries in its delta (Atchafalaya River, Pass a Loutre, South Pass, Southwest Pass), but the Missouri itself ends at the confluence.