I've come across this terminology for "linearly independent paths"
linearly independent path is any path through the program that introduces at least one new edge that is not included in any other linearly independent paths
is there a way to frame graphs as vector spaces? I can sort of see how this idea is related, but I'm having trouble seeing how to add paths in a way that makes sense and it well-defined.
They are talking about power set of the set of vertices or edges. The power set of any set is a vector space over the finite field of order $2$. The addition operation is given by taking the symmetric difference.
$$A+B=(A\setminus B)\cup (B\setminus A)$$ Scalar multiplication can really only be defined in one way.