No, the Interlinks Manager plugin doesn’t crawl your website.
How the auto internal links are applied
The auto internal links are applied with the add_filter() WordPress function after the post is retrieved from the database and before the post is printed to the screen.
The exact priority used in the filter can be defined with the Filter Priority option available in the Interlinks -> Options -> Advanced menu.
How the statistics available in the back-end are retrieved
The analysis performed on the Interlinks -> Dashboard and Interlinks -> Juice menus are delivered on demand by clicking on the “Generate Data” button.
The scripts used to generate the statistics completes their execution in few seconds and no background activities are required.