This just came up for the first time. We had two characters at Shark Island and the villain popped up at the only other open location. To temporarily close, if one character fails the check, can the other character try?
Each character there gets an opportunity to temp close, yes.
Keep in mind this only applies to temp closes. Killing a henchman that allows you to attempt a permanent close only allows whoever defeated that henchman.
Also when a location is empty, each character can try to close once during his turn.
However if you do defeat an henchman that gives an opportunity to close when defeated that happens to be the last card of a location deck, it is my understanding that you get two chances to closev the location during the same turn (one given by the henchman specific power, and one during the close location step).
Here is the relevant part of the rules that allow what you are asking:
Attempt to Temporarily Close Open Locations. When a character encounters a villain, each character at any other location may immediately attempt to fulfill the When Closing requirement for his location. If any character at a location succeeds, his location is temporarily closed and the villain cannot escape there this encounter (see Check to See Whether the Villain Escapes on page 16). Temporarily closing a location does not trigger any of the other effects of closing a location; the location opens again immediately after the encounter.
Characters may attempt to close locations in any order they wish.
So everyone gets to try, and only 1 character at a location has to succeed for it to be temporarily closed. And you can decide what order to go in. We usually let the person at the location that has the least chance of succeeding go first, if it isn't someone for whom it is impossible to fail. If they get it, great! If not, we'll use some resources (if necessary) to succeed with another character.
However if you do defeat an henchman that gives an opportunity to close when defeated that happens to be the last card of a location deck, it is my understanding that you get two chances to closev the location during the same turn (one given by the henchman specific power, and one during the close location step).
That is correct. The opportunity granted to you by the henchman is unrelated to the opportunity due to being at an empty location. So on the off chance the henchman is the last card and you defeat him, you will essentially have two chances to close the location. So try the first one with not too much cost to your resources. If you fail, you can try again but use more resources to succeed.