How to find out all the users who installed my canvas app? Is there a user interface on Facebook for this or do I need to request an API?
This cannot be done according to this - How to get all user IDs that use your Facebook app
I would advise you to use a once laid out REST API call to save all user IDs, and then add a new user to this list (save it in db).
. https://developers.facebook.com/docs/reference/fql/insights/#Metrics
application_installed_users, , .
EDIT: FQL Facebook, https://developers.facebook.com/docs/reference/fql/