Is there any way to report on words users have searched for using smart search?

You can use Campaings - you can see it works for query strings. If you set up the campaign parameter to searchtext it will work for search. You can see the searchtext query string in the URL while searching. But it will not parse the string if you search for more words.

Optionally, you could create a custom report. You may place the code (HitLogProvider.LogHit) to Search results web part.