Given the result of a dot product and one of the vectors, is it possible to find the other vector? When processing geometry the dot product or $n\cdot L$ (where $n$ is the normal, $L$ is the light direction) is baked into the $w$ of a vertex, but since the $L$ is constant I wanted to know if there is any way to get $n$ without changing the backend processing?
Given a vector $\vec v$ and scalar $a$, there is no unique solution to the equation
$$\vec v\cdot\vec w=a.$$

Consider the case $\vec v=\vec k$, and $a=0$.  Then any vector of the form
$x\vec i+y\vec j$ is a solution to 
$$\vec v\cdot \vec w=0.$$
You can't recover n from n.L and L:

Suppose v is any vector perpendicular to L, so that v.L $=0$. Then (n+v).L = n.L + v.L = n.L. So any such n+v also satisfies your condition on n.