The Live Search add-on for Firefox gives you auto-suggestions from Live Search right in the browser while you’re typing your query.
The Live Search add-on for Firefox is available to install at https://addons.mozilla.org/en-US/firefox/addon/10434. It is based on the Open Search standard.
Last year I had created a browser search box. It’s powered with Google Co-op but allows for search to be narrowed down to a specific domain.