If I have a convolution
$$z(t) = x(t) * y(t)$$
where I know $x(t)$ and $z(t)$, is there a way to determine $y(t)$? Is there a "reverse" convolution theorem for this? I know there are numerical methods used in data processing, but I'm looking for an analytical method.
There is not (generally). What you are looking for is deconvolution.
Consider the simple case where the Fourier transform of $x$ is zero somewhere. Then the other part can be arbitrary, since then:
$$ Y(\omega) = \frac{Z(\omega)}{X(\omega)} $$