The game, to be played at the War Memorial Ground, will kick-off at 7.45pm with the winners advancing to the final where they will play Nuneaton Town or Hednesford Town.
Stour have beaten Halesowen Town, Coleshill Town and Wolves to get to the semi-final stage and will now face the Baggies.
West Brom's U23 side have overcome Coventry Sphinx, Leamington and beat Alvechurch 3-2 last week to book their place in the last four.