In this case no, as live searches are tracked as events. Site search is mostly used for pageviews, when a specific page type is visited via a query argument. Usually when users visit your search page (yourdomain.com/?s=..), then it is triggered. You may have to enable it on the google analytics admin page and add the “s” as the query parameter.
We dont have a dedicated search page as we are only using the drop down search results functionality. Is it not possible to record site searches if we only have the dropdown down results option?
Yes – that should still work. It will trigger a page view as if the search results page was viewed, and after some time the data should be available in the site search dashboard.