In the state of Missouri, how long does a prosecutor have to file charges on a DWI before it is no longer valid?

Tjhe time in which a case must be filed is known as the "statute of limitations";  after that time the case is said to be "stale" (i.e. too old to file).

Each state sets its own limits, and they vary by type of offense.  Statutes of limitation on DWI / DUI prosecutions vary from state to state.  They usually permit prosecutions for at least one year after the date of the offense, and sometimes longer. If a death was involved, the DUI / DWI may also involve homicide charges, and those offenses may have very long statutes of limitations.  Also, in some circumstances statutes of limitations may be "tolled" or extended.

In MO, a felony must be charged within 3 years; a misdemeanor 1 year.

For a "simple DUI" no injuries or endangerment, a DIU is a misdemeanor.