This site is powered by Drupal for Facebook.
The page you see here is also visible in a facebook application.
I want to store the access_tokens of the users visiting my site but for some reason the fb_logins table isn't doing so. What settings enable the module to log the users to that table?
Not sure what the fb_logins table is. Probably that came from a different module.
Drupal for Facebook has a sub module called fb_user_app.module. Enable that and it will store tokens in the fb_user_app table.