Arnol'd (p. 3314) puts it that way:
Weinstein's Theorem. Some neighborhood of any Lagrangian submanifold in any symplectic manifold is symplectomorphic to some neighborhood of this Lagrangian submanifold in any other symplectic manifold, for instance in its own cotangent bundle space.
(The resulting neighborhood then has the obvious transverse polarization by fibers of the cotangent budle.bundle.) Unless I am mistaken, Weinstein proves this in Symplectic manifolds and their lagrangian submanifolds, Theorem 6.1 and Corollary 6.2 (which he points out goes back to Souriau).
