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 you a proof of the fact that any two Hamiltonian systems are locally isomorphic around non deg. points using the Moser trick (assuming of course equidimension and $h_1(x_1)=h_0(x_0)$. That is the statement I see behind the flow box theorem.

First as usal one proves the analogous linear algebraic fact which allows one then to map $(M_0,\omega_0,h_0)$ to $(M_1,\omega_1,h_1)$ in such a way that:

1. $x_0$ goes to $x_1$
2. the symplectic forms coincide on the above points and
3. the Hamiltonian $h_0$ goes to $h_1$.

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 $X_t$ under the additional restriction that $X_t$ preserves the Hamiltonian $h$. I.e. $i_{X_t}dh=0$. This means we may only choose $X_t$ in an 2n-1 dimensional subspace.

At some point in the Moser trick you have to choose a one-form $\alpha$ such that $d\alpha=\omega_1-\omega_0$, and here you have some freedom to fullfill the additional restriction. You are allowed to add any closed one form $df$ to $\alpha$ and you need the result $\alpha'$ 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 the hamiltonian $h$ 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 $Y_t(f)=g$ where $g=-i_{Y_t}\alpha$.

I hope i didn't make any big mistakes here since I just made it up.

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:

1. point $x_0$ goes to $x_1$
2. the symplectic forms coincide on the above points and
3. the Hamiltonian function $h_0$ goes to $h_1$.

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 $g_t=-i_{Y_t}\alpha$.