For short, the exponential $(X,Y)$, characterized by the usual universal properties: morphisms from any locale $Z$ to $(X,Y)$ are functions from $X \times Z$ to $Y$, exists for all $Y$ if and only if $X$ is locally compact.
The reference for this is M.Hyland's paper Function spaces in the category of locales
There is also a chapter (C4) about it in Johnstone's Elephant (Hyland paper is a bit more complet about the case of locale because it gives a description of the geometric theory of morphism that is classified by the exponential, but Johnstone also treat the case of exponential of non localic toposes.)