A simple example that shows that it doesn't always hold is an elliptically fibered $K3$ surface.
A K3 surface $S$ is a compact complex symplectic manifold of complex dimension $2$. Any smooth curve $C\subset S$ is a Lagrangian submanifold. If the Darboux Theorem were true in the sense that a neighborhood of the curve is symplectically biholomorphic with a neighborhood of the zero section of the canonical line bundle, then, for a nonsingular elliptic curve $C\subset S$, the neighborhood would be a product and so the $1$-parameter family of nearby deformations of the curve $C$ in $S$ would all be isomorphic to it, i.e., they would all have the same $j$-invariant. However, this is known not to be the case: When you have an elliptically fibered $K3$, the $j$-invariant of the elliptic fibers is not constant.
Maybe the result you want is to know that any symplectic form on a neighborhood of the zero section of the cotangent bundle that has the zero section as a Lagrangian submanifold is actually symplectomorphic to the canonical symplectic form on some neighborhood of the zero section of the cotangent bundle. This statement is true, and the standard homotopy argument proves it. (You don't need to use local coordinates for the proof, just the radial vector field on the cotangent bundle to construct the homotopy.)