According to Missouri statute, the Department of Health will issue an amended birth certificate "upon receipt of a certified copy of an order of a court of competent jurisdiction indicating the sex of an individual born in this state has been changed by surgical procedure and that such individual's name has been changed." Mo. Rev. Stat. § 193.215-9.
To apply for an amended birth certificate the applicant should submit the certified copy of the court order, a written request for the birth certificate (stating where the new certificate is to be sent), and the applicable fees to Missouri Dept. of Health and Senior Services, P.O. Box 570, Jefferson City, MO 65102.
Note that some individuals have been able to get a court order for gender change on a Missouri birth certificate without proof of surgery, as exhibited in this court order for gender change. Template forms for requesting a court order for gender change are available on the Transas City website.