Although I'm sure this isn't the point of your question, from my understanding, St. Vincent's Manhattan is not yet fully set to close. They are in talks with Continuum health partners and have several other offers to "bail out" the hospital. However, with some of the proposals, the residencies may in fact shut down and the hospital would become something like an ambulatory care center with no inpatient care.
From asking around, it's the duty of the program director of each department to make sure each resident gets placed somewhere. The residents don't just get to go anywhere they want... I'm sure there is some sort of process. However, from what I understand, many programs are happy and willing to take extra help if there are spare residents.