The proof uses the Moser technique. The normal bundle of $L$ in $M$ is $NL \cong T^*L$ (since $TM|_L = TL \oplus TL^* \cong TL \oplus T^*L$ via the symplectic form). Choose a tubular neighborhood identification $\psi: V \to U$ with $\psi|_L = \mathrm{id}$. Consider $\omega_0 = \omega_{\text{can}}$ and $\omega_1 = \psi^*\omega$ on $V$. Both restrict to zero on $L$ (Lagrangian condition) and are non-degenerate near $L$. The convex combination $\omega_t = (1-t)\omega_0 + t\omega_1$ is non-degenerate near $L$ for all $t$. By the relative Moser argument (fixing $L$), there exists a diffeomorphism $\phi: V \to V$ with $\phi|_L = \mathrm{id}$ and $\phi^*\omega_1 = \omega_0$. The composition $\psi \circ \phi^{-1}$ gives the desired symplectomorphism. $\square$