There is an entire department at the Academy that is dedicated to the Sea Year (it is a rather small one though). When it comes time for the midshipmen to go to sea they will set them up with their ship and make all the travel arrangements. There can be some small hiccups and it is necessary for the midshipmen to have a credit card in case something unplanned happens.
Basically they will get the midshipmen tickets to the ship and back to the Academy. Other expenses, such as cab fare or unexpected hotel stay, the midshipman will pay for and then be reimbursed upon return to school at the end of sea year.
The main sticking point that many people run in to is that the Academy only furnishes travel to and from the Academy. They will not reimburse for travel from the ship to home and then back to the Academy if the midshipman has some time off at the end of sea year.
You would probably have to pay the bill before your midshipman gets back from sea because it wouldn't be possible to get reimbursed that quickly, but it shouldn't be too much of a burden. Once they get back to the Academy and are reimbursed it would be up to you to ask for the money from your midshipmen or let them have it as a little extra spending money