I'm hoping for unique solutions for a nonlinear system of the following form:
$x' = f(x,y,z)$,
$y' = g(x,y,z)$,
$z' = h(x,y,z)$.
I know with a single equation, finding the Lipschitz constant would be helpful in proving uniqueness. Is there something analogous for a system?
Yes, and the answer is exactly the same. If we write $w=(x,y,z)$ and $F(w)=(f(w),g(w),h(w))$ (so we combine the three dependent variables into a vector and the nonlinearity into a vector field), then the Lipschitz condition for $F$ reads $$\|F(w_1)-F(w_2)\|\le K \|w_1-w_2\|,$$ where $\|\cdot\|$ is some norm on vectors, for example $\|(x,y,z)\|=\sqrt{x^2+y^2+z^2}$.
A Lipschitz condition of this type implies existence and uniqueness of solutions for systems of ODEs exactly as in the single equation case.