Every captain, beginning with the next one after you connect the account, receives a chest, regardless of what previous captains have done with it, including finishing the related quest or throwing it in the waves.

The chest shows up as an item that does not take up space in your hold, and possession of it allows you to take it to the Alarming Scholar in London to begin the quest to open it, though at many points during the quest you can also sell it, and you can also attempt to force it open or sacrifice it to Salt at any time from your hold. The quest is not an Ambition, nor does it replace one.

The "next Captain" line is a very poorly worded method of indicating that you won't receive it until you die, if you have a game already in progress.