If there is no tracker, you'll have to create a new one - by clicking on the "New [Eureka] Tracker".

That tracker will have an ID that you can share with others then.

The website interacts in no way with the game - you'll have to check what FATEs are killed manually. Thats why you wont get a tracker ID from the game either - the game itself has no Eureka-trackers. This website is only a tool to check off what NMs have been killed and to share it easier, each tracker gets an ID.