I don't think there is a correspondence between simplicial spaces and certain internal locales in 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.

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.