$\forall y \in \mathbb{R}, \int_{-\infty}^{\infty} f(x)f(x-y)dx=f(y)$
I also know that $\int_{-\infty}^\infty f(x) dx$ converges and that $f$ is symmetric about the origin. What does $f$ look like? Is it possible to identify a parametric set of solutions for $f$? How do you even go about solving a problem like this?
A Fourier transformation leads to $\hat f^2=\hat f$, which is solved by any Fourier transform $\hat f$ that takes only the values $0$ and $1$. For instance, $\hat f=\chi_{[-a,a]}$, the characteristic function on an interval centred on the origin, leads to a $\operatorname{sinc}$-like function $f$.