Can the algebraic geometry of schemes be developed internally in topoi?

The notes of Ingo Blechschmidt, Using the internal language of toposes in algebraic geometry cover this. See also his lecture at Topos à l'IHÉS of the same name.

His work very much extends Hakim's thesis.

Perhaps you might find some answers in Monique Hakim's book, Topos annelés et schémas relatifs. I only looked at it briefly long ago, so I can't be sure if it's relevant, but Mathreviews quotes part of the introduction "... Au Chapitre IV on définit la catégorie SchS des schémas relatifs sur un topos annele S ..." So maybe it is.