No, the Local Search Rank checker report is not run through your browser or IP address.
We use a huge pool of IPs on proxy servers, located in hundreds of different towns and cities throughout the countries in which we operate.
The IPs that we use are intelligently rotated in order to keep them from being detected and blocked by search engines. Your own IP won't be flagged by search engines, and it doesn't consume your personal bandwidth.
This approach also avoids complications of 'personalization' that skew search engine results based on your search history and websites you've visited.