The father is not the biological father, but the son is a sperm donor son agreed upon by both parents. If they get a divorce in Louisiana can she leave the state with her son as she is the blood parent?

If the father is the legal parent then he has rights if he chooses to let mother and child go that's up to him.

Agreed, not sure what you mean by "sperm donor" Did she go to a sperm bank ?

Is he the legal father, is he the actual biological father ( sperm donor) or one or more?

But if he is the legal father, he will have rights , if he chooses to use them.