What is a legal cause for emancipation in Florida?

There is no 'legal cause' for emancipation, but if there are reasons you feel should be considered by the court, you put them in the petition. You have to meet the requirements of the Florida statute:

The petition shall contain the following information: (a) The name, address, residence, and date of birth of the minor. (b) The name, address, and current location of each of the minor's parents, if known. (c) The name, date of birth, custody, and location of any children born to the minor. (d) A statement of the minor's character, habits, education, income, and mental capacity for business, and an explanation of how the needs of the minor with respect to food, shelter, clothing, medical care, and other necessities will be met. (e) Whether the minor is a party to or the subject of a pending judicial proceeding in this state or any other jurisdiction, or the subject of a judicial order of any description issued in connection with such pending judicial proceeding. (f) A statement of the reason why the court should remove the disabilities of nonage.