Organization pages already limit searches to the current organization:

User profiles should behave the same:

This issue replaces #548
2018 edit: this type of search has been dropped from GitHub, this might not be easily possible
Reopening.
The code this feature used to rely on was dropped in the last GitHub update and the label was left non-functional:

@bfred-it It's unclear what the actionable thing to do here is?
Restoring the feature, somehow. This is on your profile:

"Search this user" could appear up there, like (mockup):

This appears when searching from a repo, so multiple actions should already be possible:

That makes sense. 馃憤
Hey hey 馃憢
So I wanted to give you a heads up that this is something that the Papercuts team at GitHub noticed and are actively working on. Thought it best to let you know!
Most helpful comment
Hey hey 馃憢
So I wanted to give you a heads up that this is something that the Papercuts team at GitHub noticed and are actively working on. Thought it best to let you know!