Yes, FW have their own business. They may even be the head of their own house and business, before their brothers.
The lady Constanzia of Besnit, in Witness, is a good example. She led her wealthy merchant house, she was first in the house, despite she had brothers.
There are examples of FW, in books, who are totally independent and don't rely on a protector, guardian, male family members or free companion.