It appears that Microsoft have made some recent changes to the latest Outlook builds of Outlook 2013 onwards, and these changes are causing some users problems with normal Outlook search functionality under certain circumstances.
This is not an issue with the hosted Exchange service itself. Searches of your mailbox from within Outlook on the Web still occur normally.
This guide will help you work around the issue.
Firstly, close Outlook.
Now download the following file, save it to your Desktop and unzip it, then double-click the .reg file and select OK when asked if you want to add entries to your Windows registry.
Disable Server Assisted Search (right mouse click then "Save link as..." to save the .zip file)
You can now restart Outlook and any searches performed will be against your locally cached OST file, as they should be.
The exception is for native Exchange archive mailboxes, which will be searched in real time as normal.