I'm wanting the information in the "Inbox" tab of the notifications in the top left of any SE website. How can I get this or is it not yet available?
no - the 'super-collider' is not exposed by the API.
you would need to synthesize this by maintaining a last visit data for a user and then polling each site's api endpoint for several queries filtered on last_activity or created_date.
this is possible but you would really need to be commited to getting the results.