Actually, it's easy to solve this problem for unbounded polygonal domain whose boundary is connected.
Let $\Omega$ be such a domain, and let $w = u+iv:\Omega\to\mathbb{C}$ be a Schwarz-Christoffel transformation that maps $\Omega$ one-to-one onto the upper half plane and sends the point at infinity to the point at infinity. Then $v$ vanishes on the boundary of $\Omega$ and is positive on the interior of $\Omega$. Let $g = \mathrm{d}u^2 + \mathrm{d}v^2$, which is a positive definite form on the interior of $\Omega$ (in fact, conformal to the standard metric on $\Omega$).
Let $X=-\partial/\partial v$. This solves the problem, since $X$ is perpendicular to the boundary of $\Omega$ and it clearly has divergence zero and length $1$ with respect to $g$.
(Of course, $X$ is not continuous at the corner points.)