From what I can tell, Godot has limited support for non-BMP Unicode characters, but I have no idea how that support extends to JSON. I know that there are plans to improve the Unicode support in Godot with Godot 4, so maybe it will be included then? I wasn't able to find anything JSON specific though.
If there is a use case for it and you think it would be a good addition, I would suggest opening a proposal on the Godot Proposal repository.
I don't know enough about non-BMP unicode to say whether the feature could/should be implemented, though if Godot's other text-based stuff (labels, GDScript, etc) support non-BMP unicode, then I think it is reasonable to have JSON support it too.