Actually, there are two speakers during the ending; the first half of the narration can be attributed to Mikoto, the second half to Vivi.

And, yes, Vivi does die. Vivi's final lines...

I'm so happy I met everyone... I wish we could've gone on more adventures. But I guess we all have to say goodbye someday. Everyone... Thank you. Farewell. My memories will be part of the sky...
...hint directly at the fact that he has stopped functioning; the critical fact that he does not appear during the final play scene, even though his "sons" are present, also attests to this. The inevitability of Vivi's death is further suggested by a conversation with Black Mage No. 288, whose response to Vivi's hesitant question concerning "stopping" is shown as extended ellipses.

Exactly how Vivi's "sons" came to be is not revealed during the ending, so one can only speculate on how his progeny came about.