I don't think there is a correspondence between simplicial spaces and certain internal locales in $\mathbf{sSet}$.
Constructing an internal locale is the same thing as defining an internal frame, only the notion of morphism is different. An internal frame in $\mathbf{sSet}$ is a functor $\Delta^\mathrm{op} \to \mathbf{Frm}$ satisfying certain conditions (see Sketches of an Elephant, C1.6, Lemma 1.6.9). Here $\mathbf{Frm}$ is the category of frames. On the other hand, a simplicial space is a functor $\Delta^\mathrm{op} \to \mathbf{Top}$, with $\mathbf{Top}$ the category of topological spaces. You can associate with a topological space its frame of open sets, but then you get a functor $\Delta^\mathrm{op} \to \mathbf{Frm}^\mathrm{op}$ instead of a functor $\Delta^\mathrm{op} \to \mathbf{Frm}$.
On the other hand, what you can do is look at the topos $[\Delta, \mathbf{Set}]$ rather than $\mathbf{sSet} = [\Delta^\mathrm{op},\mathbf{Set}]$. Then each internal locale in $[\Delta, \mathbf{Set}]$ is given by a functor $\Delta \to \mathbf{Frm}$. If you are lucky, then all frames in the image of this functor are spatial, and then you can associate to this internal locale a functor $\Delta^\mathrm{op} \to \mathbf{Top}$, or in other words a simplicial space. It would be interesting to see which simplicial spaces come from internal locales in $[\Delta,\mathbf{Set}]$, but as far as I know this is very difficult.
