The closest analog I am aware of to the Weinstein model for Lagrangian submanifolds in the contact setting is the so-called "Lychagin chart:" Every Legendrian submanifold $L$ is contactomorphic to a neighborhood of the zero section of the space of one-jets on $L$. See Banyaga, "The Structure of Classical Diffeomorphism Groups," Section 6.2, also Lychagin, "Local classification of nonlinear first-order partial differential equations," Russ. Math. Surv. 39 (1975), 105-175.

The closest analog I am aware of to the Weinstein model for Lagrangian submanifolds in the contact setting is the so-called "Lychagin chart:" Every Legendrian submanifold $L$ has a neighborhood contactomorphic to a neighborhood of the zero section of the space of one-jets on $L$. See Banyaga, "The Structure of Classical Diffeomorphism Groups," Section 6.2, also Lychagin, "Local classification of nonlinear first-order partial differential equations," Russ. Math. Surv. 39 (1975), 105-175.