There is no way, free or otherwise, to find out exactly who performs a specific search online. The reputation management firm BrandYourself offers partial data about some searches as part of its free service, but does not provide names and does not cover every search.
BrandYourself creates a profile for the user, which is intended to rank highly in search engine results. When someone searches for the user's name and visits this profile, BrandYourself logs the visitor's IP address. This allows them to report general geographic locations of searches. It does not offer any personally identifiable information, and it only tracks visitors who click the profile link.