Suppose $h\in S(\mathbb{R}^d)$ (Schwartz space) and a family $\mathcal{F}=\{f(\,\cdot\,;s)\}_{s\in\mathbb{R}^d}\subseteq S^\prime(\mathbb{R}^d)$ of tempered distributions. Then, for each fixed $s$ we have $f(h;s)\in\mathbb{R}$.
Now, assume that the map $s\mapsto f(h;s)$ is again in $S(\mathbb{R}^d)$ and take $g\in S^\prime(\mathbb{R}^d)$. Denote the action of $g$ over $s\mapsto f(h;s)$ by $\langle g_\cdot,f(h;\,\cdot\,)\rangle$.
Is there some theorem/propertie that guarantees that there is a continuous function $M\{g,\mathcal{F}\}:\mathbb{R}^d\to\mathbb{R}$ such that for all $h\in S(\mathbb{R}^d)$ we have $$\langle g_\cdot,f(h;\,\cdot\,)\rangle=\int_{\mathbb{R}^d}h(s)M\{g,\mathcal{F}\}(s)\,\mathrm{d}s\,?$$
EDIT Boris'answer is correct. I'll change the question:
Is there some tempered distribution $M\{g,\mathcal{F}\}$ such that $\langle g_\cdot,f(h;\,\cdot\,)\rangle=M\{g,\mathbb{F}\}(h)$?
In Boris answer, the tempered distribution $M\{g,\mathbb{F}\}$ is $g$.
Let $f(\cdot;s)=\delta(x-s)$ and $g=\delta(x)$, then $f(h;\cdot)=h(\cdot)$ and $\langle g_\cdot,f(h;\cdot)\rangle=h(0)$, so there is no such function, that $$ h(0)=\int_{\mathbb{R}^d}h(s)M\{g,\mathcal{F}\}(s)\mathrm{d}s $$