If it's a seat washer that needs replacing, then that's what you buy. Either flat or conical. It might also be some cheesy import sillcock that used an O-ring instead of a washer. Kind of hard to make out from over here. In any event, there should be an indoor shutoff in the line feeding the sillcock. Of course, the very expensive kind of 'through-the-wall hydrants' you see on commercial buildings can have their own proprietary rubber parts, and you won't find replacements to be had short of special-ordering them through a plumbing supply.