In ArcMap, the user clicks the Search button and nothing happens (no search window opens or appears)
This can happen if ArcMap fails unexpectedly and the search window location stored in the user's profile gets corrupted, or sometimes when the user moved the search window to another display and that display is no longer connected. (The search window may come up, but who knows at what screen address!)
Here are a few fixes to try, in this order. Close ArcMap, try the fix, and restart ArcMap and see if it fixed the problem.