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Universes seem to first appear in Grothendieck's work in SGA 1, which is credited to Grothendieck, and a lengthy discussion is in the chapter on Prefaisceaux (presheaves) in SGA 4. That chapter is credited to Grothendieck and Verdier. The appendix on them there is credited to N Bourbaki.

Is there any known evidence of who actually wrote the appendix?

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I have no solid evidence, but I believe it was written by Grothendieck himself. –  Joël Oct 20 '13 at 6:17
    
I believe that too, based the bits of set theory in other things he wrote, as already in the Tohoku paper. I fear there is direct evidence in some long-unread letters or notes somewhere that I will never find. –  Colin McLarty Oct 21 '13 at 0:57

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