Who has authority to close a runway/airport?
Generally two groups have authority to close a runway (or airport) for an incident/emergency, and if either says it's closed, it's closed:


The airport owner/operator/management.
It's their facility, so if they say it's closed, it's closed.
Specifically it's the persons authorized to issue NOTAMs for the airport who can close a runway (by issuing the NOTAM saying it's closed), but the chain of command for making this decision may vary from airport to airport.
Runways may also be closed by airport administration without issuing a NOTAM (this happens frequently at Brookhaven airport (KHWV) - "Closed Runway" X's will be put out on the crosswind runway when the banner tow operators are set up for pick-up/release on that runway). Often there's a blanket NOTAM about "banner tow ops" or similar that sort-of covers this situation (basically "Expect strange things to be happening because the banner tow people are doing...whatever it is they do".)
The control tower (if one exists).
Whether or not they have the administrative authority to "close" a runway, they certainly have the practical authority to do so by denying anyone clearance to take off/land from it or to enter it, and by issuing NOTAMs indicating that a runway is closed/unusable for some reason.


At an airport like LaGuardia where you have both an FAA Control Tower and a robust airport operations/management group (The Port Authority of New York &amp; New Jersey) a Memorandum of Understanding (MOU) or similar agreement is likely in place to ensure that everyone knows who is authorized to do what when, but in cases like this where an aircraft has a runway excursion the course of action is pretty obvious and nobody is going to want aircraft to be moving around on the ground until the situation is under control and safety can be assured.