What is "legal authority" which is required for iOS developer program?
Apple's phrasing is deliberately wide because the terms need to cover a wide range of countries and legal systems.

Depending on the jurisdiction you are in, the meaning of legal authority will differ. Typically, it means the individual agreeing has the power to make a binding agreement between the individual or organisation and Apple Inc.

As an example, a Limited Liability company registered in England and Wales would typically require a Director to make the agreement. The Director in this case being able to represent the company unlike an employee or sub-contractor.

As always, if you are unsure of your legal role or position, please get professional advice.