Let $(M, \omega)$ be a symplectic manifold and $Q \subset M$ a coisotropic submanifold of codimension $k$. I'm trying to prove that for every $x \in Q$, there exists an open subset $U \subset M$ containing $x$ and a submersion $F : U \to \mathbb{R}^k$ whose coordinates are in involution (i.e. $\{F_i, F_j\} = 0$) such that $F^{-1}(0) = Q \cap U$.
I proved this in the case $Q$ is Lagrangian, as a corollary of the fact that $Q$ would have a neighborhood symplectomorphic to a neighborhood of the $0$ section in $T^{*}Q$, which would allow us to get "adapted" Darboux coordinates: $(x_1, \ldots, x_n, y_1, \ldots, y_n)$ Darboux coordinates around $x$ where $Q$ is described by $y_1 = \cdots = y_n = 0$. Then we can just let $F(x_1, \ldots, x_n, y_1, \ldots, y_n) = (y_1, \ldots, y_n)$ and it works.
In the general coisotropic case, I can't figure out what to do. It seems likely one should be able to get Darboux coordinates where $Q$ is determined by $y_{n - k + 1} = \cdots = y_n = 0$, but I can't see how.