Denote $F(x)=G(x)e^{x^2/2}$, this multiple is chosen in order to get an equation of the form $G(x+a)-G(x-a)=$ a given function. Indeed, you have $G(x)\in [0,e^{-x^2/2}]$ and $$G(x-a)e^{x^2/2+a^2/2}+e^{-xa}-G(x+a)e^{x^2/2+a^2/2}=e^{a^2/2}\\ G(x+a)-G(x-a)=e^{-(x+a)^2/2}-e^{-x^2/2}.$$ One such function $G=:G_0$ is straightforward: $$G_0(t)=\sum_{k=0}^\infty (-1)^ke^{-(t-ka)^2/2},$$ then we are given that $G-G_0$ is $2a$-periodic. On the other hand, $G(x)$ goes to 0 when $x$ goes to $-\infty$, and so does $G_0$. Thus $G-G_0$ is a periodic function which goes to 0 when $x$ goes to $-\infty$. The only such function is identical 0. Therefore $G\equiv G_0$, and the question reduces to whether $G_0(x)\in [0,e^{-x^2/2}]$ for all $x$. But $$\lim_{n\to +\infty}G_0(x+2na)=\sum_{k\in \mathbb{Z}}(-1)^k e^{-(x-ka)^2/2}$$ depends on $x$ (it is closely related to Jacobi theta-function) and is not in general equal to 0), so alas, your $F$ does not exist.

Denote $F(x)=G(x)e^{x^2/2}$, this multiple is chosen in order to get an equation of the form $G(x+a)-G(x-a)=$ a given function. Indeed, you have $G(x)\in [0,e^{-x^2/2}]$ and $$G(x-a)e^{x^2/2+a^2/2}+e^{-xa}-G(x+a)e^{x^2/2+a^2/2}=e^{a^2/2}\\ G(x+a)-G(x-a)=e^{-(x+a)^2/2}-e^{-x^2/2}.$$ One such function $G=:G_0$ is straightforward: $$G_0(t)=\sum_{k=0}^\infty (-1)^ke^{-(t-ka)^2/2},$$ then we are given that $G-G_0$ is $2a$-periodic. On the other hand, $G(x)$ goes to 0 when $x$ goes to $-\infty$, and so does $G_0$. Thus $G-G_0$ is a periodic function which goes to 0 when $x$ goes to $-\infty$. The only such function is identical 0. Therefore $G\equiv G_0$, and the question reduces to whether $G_0(x)\in [0,e^{-x^2/2}]$ for all $x$. But $$\lim_{n\to +\infty}G_0(x+2na)=\sum_{k\in \mathbb{Z}}(-1)^k e^{-(x-ka)^2/2}$$ depends on $x$ (it is closely related to Jacobi theta-function) and is not in general equal to 0, so alas, your $F$ does not exist.