Missouri law lets private employers adopt a veterans preference policy for hiring and promotion. The policy can cover veterans, the spouse of a disabled veteran with a service-connected total and permanent disability, and the surviving spouse of a deceased veteran.
There is nothing to apply for with the state. Ask an employer whether they have a veterans preference policy and how to claim it.
Your DD214, or your marriage certificate plus the veteran's VA disability or death documentation, if the employer asks for proof.