Contrary to the OP's expectation, the integro-differential equation - henceforth referred to as (#) - can be solved explicitly. It is natural to prescribe the value of $f'(0)$ which is the boundary condition I'll consider below. As an effect of the nonlinearity, one then has two, one, or zero solutions.

It is convenient to work with the parameter $\kappa>0$, where $\sigma^2/2 = (\kappa+1)/\kappa^2$ (i.e., $\kappa=\left(1+\sqrt{1+2\sigma^2}\right)/\sigma^2$).

The result is as follows:

All solutions $f$ of (#) are of the form
$$
f(x) = c - d e^{-\kappa x},
$$
where $c\geq0$ and $d\in\mathbb R$ are subject to the condition
$$
\Phi(d) = \Psi(c) \tag{*}
$$
with $\Phi$ (depending on $a$ and $\kappa$) and $\Psi$ being entire functions (see below). If $f$ is a solution , then $c$ is the value of the integral in the right-hand side of (#).

There are three critical values $a_c$ (depending on $\kappa$), $0$, and $2\kappa$ for $a$ such that the following holds:

For $a\leq a_c$, there is a $d^-<0$ (depending on $a$ and $\kappa$) such that (#) is solvable if and only if $f'(0)/\kappa\in[d^-,\infty)$.

For $a_c<a<0$, there are $d^-<0<d^+<d^{++}$ such that (#) is solvable if and only if $f'(0)/\kappa\in[d^-,d^+]\cup [d^{++}, \infty)$

For $0\leq a<2\kappa$, there are $d^-<0<d^+$ such that (#) is solvable if and only if $f'(0)/\kappa\in[d^-,d^+]$.

For $2\kappa\leq a$, (#) is unsolvable.

**Note.** $a_c<0$ is implicitly determined by the condition that the (overdetermined) system $\Phi(d)=e^{-1}$ and $\Phi'(d)=0$
has a solution $d>0$.

**Proof** (sketch)**.**

First one has to solve the ordinary differential equation $f(x) - \frac{\kappa+1}{\kappa^2}\,f''(x) -f'(x)= c$ on $[0,\infty)$ for a constant $c\geq0$. The bounded solution with $f'(0)/\kappa =d$ is $f(x) = c - d\,e^{-\kappa x}$. Then one is left with the integral equation
$$
\kappa^2 d^2 e^c \int_0^\infty e^{-d\,e^{-\kappa x}+ax-2\kappa x}\,dx = c.
$$
The integral in the left-hand side converges precisely when $a<2\kappa$ and can be done explicitly. The result is (substitute $y= d\,e^{-\kappa x}$ if $d>0$, then use analytic continuation):
$$
\underbrace{\kappa\,d^{\,a/\kappa}\,\gamma\left(2-\frac{a}\kappa,d\right)}_{\Phi(d)} = \underbrace{c\,e^{-c}}_{\Psi(c)},
$$
where $\gamma(s,x) = \int_0^x y^{s-1} e^{-y}\,dy$ is the lower incomplete Gamma function. Note that the function $x^{-s}\,\gamma(s,x)$ for $s>0$ (in our case $s=2-a/\kappa$) equals $\int_0^1 y^{s-1}e^{-x y}\,dy$ which is an entire function of $x$.

I'll only discuss the case $0<a<2\kappa$. In this case, one can show that $\Phi$ is strictly decreasing from $\infty$ to $0$ on the interval $(-\infty,0]$ and strictly increasing from $0$ to $\infty$ on the interval $[0,\infty)$, while $\Psi$ is strictly increasing from $0$ to $e^{-1}$ on the interval $[0,1]$ and strictly decreasing from $e^{-1}$ to $0$ on the interval $[1,\infty)$. Denoting by $d^-< 0 < d^+$ the two solutions $d$ of the equation $\Phi(d) = e^{-1}$, then there arise the following four cases:

- for $d=0$ one has $c=0$,
- for $d=d^-$ or $d=d^+$ one has $c=1$,
- for $d\in (d^-,0)\cup (0,d^+)$ there are two solutions $c$ of (*),
- for $d\in (-\infty,d^-)\cup (d^+,\infty)$ there is no solution $c$ of (*).

**Example.** For $a=1$, $\kappa=1$ (i.e., $\sigma=2$), (*) becomes
$d\left(1-e^{-d}\right) =c \,e^{-c}$. Here is a plot of this relation:

($d^-=-0.528399$ and $d^+=0.718078$.)