The hamiltonian flow box theorem, as stated in Abraham and Marsden's Foundations of Mechanics, says that:
Given an hamiltonian system $(M,\omega,h)$ with $dh(x_0)\neq 0$ for some $x_0$ in $M$, there is a symplectic chart $(U,\phi)$ on $M$ centered at $x_0$ such that $\phi_{\ast}h(x)=h(x_0)+\omega_0(\phi_{\ast}X_h(x_0),x)$, where $\omega_0$ is the canonical symplectic form.
*Question:*I know some different proofs of this theorem, but I would know if, at your knowledge, in the literature there is a proof which uses the Moser's trick as in the proof of the Darboux' theorem.
In Abraham and Marsden there is a proof using the contact structure associated to the symplectic one and its canonical transormations.
I know even that it has an extension in a theorem of Cartan which says:
Given a $2n$-dimensional symplectic manifold $(M,\omega)$, it is possible to extend to a system of symplectic coordinates on $(M,\omega)$ any set of local functions $f_1,\ldots,f_k,g_1,\ldots,g_l$ on $M$ such that $f_1,\ldots,f_k$ are independent and in involution, $g_1,\ldots,g_l$ are independent and in involution, and $\{f_i,g_j\}=\delta_{ij}$ for any $i,j$.
Best Answer
My guess would be that you can find such a proof in the literature, since the Moser trick is such a powerful tool, though I don't know where.
Instead let me sketch a proof of the fact that any two Hamiltonian systems $(M_i,\omega_i,h_i)$ are locally isomorphic around non deg. points $x_i\in M_i, i=0,1$ using the Moser trick. That's the statement I see behind the flow box theorem.
First as usal one proves the linearized fact (i.e. any two $2n$ dim vector spaces $V_i$ supplied with non-degenerate two-forms $\omega_i$ and non degenerate one forms $v_i, i=0,1$ are isomorphic). Using this one constructs a local diffeomorphism between $(M_0,\omega_0,h_0)$ and $(M_1,\omega_1,h_1)$ satisfying:
Now we have a manifold $M$ with two symplectic forms $\omega_0,\omega_1$ coinciding at $x\in M$ and one function $h$ which is nondegenerate at $x$. Next you try the usual Moser trick to morph $\omega_1$ to $\omega_2$ with the flow of a time dependent vector field $X_t$, imposing the additional requirement that $X_t$ preserve the function $h$. I.e. $L_{X_t}h=0$. Hence $X_t$ should lie in the $2n-1$ dimensional distribution $\ker dh$.
At some point in the Moser trick one chooses a one-form $\alpha$ such that $d\alpha=\omega_1-\omega_0$, and here we have the freedom to fullfill the additional restriction since we can add any closed one form $df$ to $\alpha$. We want the result $\alpha+df$ to lie in the $2n-1$ dimensional subspace of one-forms satisfyinig $i_{Y_t}\alpha'=0$, where $Y_t$ denotes the Hamiltonian vector field associated to $h$ w.r.t. the symplectic structure $\omega_t$. This can always be achieved since $Y_t$ is non deg and you are solving the equation $Y_t(f)=g_t$ where $g_t=-i_{Y_t}\alpha$.