Yes, the whole family has the ability. This is made pretty clear in the book (not so sure about the TV series/radio series/comics).
For example, Portico's "house" consist of a series of rooms spread out all over the world, linked in a way that only a member of his family could go from one to the other. The Marquis comments in astonishment upon this as an excellent security setup, and can't figure out how Croup and Vandemar managed to defeat it (which we never do find out exactly).
Door also mentions casually to Richard that her family has a certain gift with doors.
If I recall correctly, there's also a bit towards the end where the heroes are informed that they can kill Door at need because they still have Door's little brother. Although I believe that turned out to be a lie, the implication is that they have another person able to open doors at need.
I think there is also a bit near the beginning where Door is being chased and tries to calm herself by going through exercises that her father taught her in order to Open.
It's also symbolic with her father having plans to unite London Above and London Below. That he would be the one to open the door to doing so.
I think the Earl also comments at one point about Portico's family all having that funny thing with doors.