"She is a member of this organization."
Translation:Ela é um membro desta organização.
Would it always be "o membro" even if referring to a female?
Yes, it is incorrect.
By membro da organização it means that someone is part of such an organization. The rank and file has not been specified.
When meaning affiliated or associate you can use afiliado or associado.
The most common meaning of sócio or sócia refers to someone who owns a business and shares its profits with its peer (business partner).
You can also use sócio or sócia out of the business context. It could be used when you are in an endeavor with someone else in which your peer is sharing the efforts to do something with you (as well as the results of such an endeavor).