We are all adults in the US at the age of 18 and are free to live where we wish.
There can be exceptions. A person's freedom to go and live where they choose can be limited if they pose a safety risk to themselves or others, are deemed mentally incompetent (unable to make decisions for themselves and/or care for themselves), or if they are convicted of some crimes (jail or probation). YOu mention she had legal trouble. Is she in trouble for leaving the home, or for some crime she committed? And is she in trouble for leaving Girls Town or for avoiding a court date or violating probation terms?
The age of majority is not 18 in all States - there are exceptions where it is 19 or 21. Missouri is NOT one of the exceptions.
A person's freedom can only be limited if a Court decides it is necessary and appropriate.
This facility - Girls and Boys Town of Missouri - is not a foster care facility; it is described as follows: "Children must be referred to Boys & Girls Town of Missouri by the Department of Social Services Children's Division, the Department of Mental Health, the Division of Youth Services, the juvenile court system, private insurance companies, and parents. It would appear that OP's sister had to be referred to the facility, very possibly by the Court, and that is why they are looking for her after she left without permission.
This is - to a large extent - a mental health facility geared toward children and young adults.