Such information is unlikely to be publicly accessible, for privacy reasons. Vital records are generally not public records. Missouri state regulations (19 CSR 10-10.090) dictate that only certain people may have access to birth certificates, and the information contained therein:
As authorized by section 193.255.1, RSMo 1986, the registrant, a
member of his/her immediate family, his/her guardian or one (1) of
their respective official representatives shall be considered to have
a direct and tangible interest and may be issued a certified copy of a
vital record. Others may demonstrate a direct and tangible interest
when information is needed for determination or protection of personal
or property rights.
As a genealogist who is not an immediate family member, you have no rights to access the birth certificate, or the information contained in that birth certificate.
This being said, there may be ways to find out this information. Try asking your relatives directly. Many will be more than happy to provide you with basic family details. You can also explore social media. It is somewhat frightening how much information people leave publicly accessible in their Facebook profiles.
Finally, ask yourself why you need this information? Is it just to fill a gap in your family tree? Family history starts in the present, but keep in mind that some people – whether right or wrong – may not take kindly to having their children's information in your family tree. If you do obtain this information, be careful not to reveal it online or in some other public forum without obtaining their permission.