  Weyl group multiple Dirichlet series are Dirichlet series in $r$ complex
variables, with analytic continuation to $\mathbb{C}^r$ and a group of
functional equations isomorphic to the Weyl group of a reduced root system of
rank $r$. Such series may be defined for any global field $K$, but in the case
when $K$ is an algebraic function field they are expected to be, up to a
variable change, rational functions in several variables. We verify the
rationality of these functions in the case when $K=\mathbb{F}_q(T)$, and
describe the denominators and support of the numerators.
