Written permission or prohibition to bring ham radio gear on board is proffered by some of the large cruise lines. For example, Carnival gives written permission by way of their published policy where as Norwegian specifically denies permission to possess ham radio equipment on board (or even a satellite phone!).
Once on board, the captain of the ship must be approached to garner permission to operate. You can expect to be asked about the antenna, the frequencies involved, the amount of power, and how the radio will be powered. The captain will view most of this information with an eye toward the safety of fellow passengers and the potential for interference with on-board radio equipment. If the captain is so inclined he or she will then give verbal permission but most likely with the caveat that at his or her request, you must immediately cease operations. The captain most certainly will not take the time during boarding to grant you any type of written permission - everything will be verbal.
So unless your country's regulations specifically require that you obtain written permission, you should be able to operate with the captain's verbal permission. Do keep in mind that your home country's radio regulations may only be applicable when you are within your country's own territorial waters. If you are in international waters, often the flag of the ship determines the applicable regulations for the ham radio operator.