The correct answer is B) refer to the vessel's Certificate of Inspection. The Certificate of Inspection (COI) issued by the US Coast Guard specifies the types and grades of cargo that a tank vessel is authorized to carry. This document contains the official record of the vessel's design, equipment, and operational limitations, including the cargo the vessel is permitted to transport. Referring to the COI is the appropriate way to determine the authorized cargo grades for a tank vessel, as it is the definitive regulatory document. The other options are incorrect because: A) the terminal supervisor does not have the authority to override the vessel's COI; C) inspecting the cargo tanks and fittings alone does not provide the legal authorization for cargo carriage; and D) checking the loading order does not directly indicate the permitted cargo grades.