Is there a semantics for intuitionistic logic that is meta-theoretically "self-hosting"?

I would argue that intuitionistic logic is perfectly self-hosting: working in an intuitionistic set theory, one can define a sound semantics of intuitionistic logic relative to models built out of plain sets in the (intuitionistic) metatheory, without any need for Kripke-ness or anything complicated. Just as the interpretation by set-models in a classical metatheory is the "intended" meaning of classical logic, a word-for-word interpretation by set-models in an intuitionistic metatheory is the intended meaning (or, at least, an intended meaning) of intuitionistic logic.

What forms of completeness theorem are intuitionistically provable is a separate question, which I would argue doesn't impact on the self-hostingness of intuitionistic logic. But I believe the forms of completeness that are true for intuitionistic logic — which are arguably best expressed in terms of models in arbitrary categories or hyperdoctrines of the appropriate sort, including Kripke models as the special case of presheaf categories — can certainly be proven in an intuitionistic metatheory. For instance, Part D of Johnstone's Sketches of an Elephant proves these categorical completeness results for various fragments of logic, including full first-order intuitionistic logic, and generally uses intutionistically valid reasoning.

Comment environment was acting funny, so I am writing an "answer". Here is a good place to start:

Harry de Swart's PhD from the University of Nijmegen (the Netherlands) was about this kind of topic:

H.C.M. de Swart: Intuitionistic logic in intuitionistic metamathematics, Dissertation, 1976, University of Nijmegen. https://core.ac.uk/reader/43594080

He has since left this field, so I do not know if he will answers questions via email.

See Palmgren, Constructive Sheaf Semantics for a completeness proof for sheaf semantics within a constructive (and predicative) metatheory. The introduction also mentions several references to earlier approaches that are generalised by sheaf semantics. I recommend particularly Troelstra and Van Dalen, Constructivism in mathematics, vol 2 as a standard reference containing various results about the semantics of constructive mathematics.