We tackle the issue of scalability when synthesizing controllers for signalized vehicular traffic networks with linear temporal logic specifications. Vehicular traffic networks are natural candidates for a compositional synthesis approach because they are naturally decomposed into sub-networks. However, naively synthesizing controllers for individual sub-networks and interconnecting them can violate the specifications on the monolithic network. Through notions of supply and demand in our system dynamics, we construct contracts between sub-networks that guarantee the soundness of the overall synthesized controller. Furthermore when the network exhibits monotone dynamics and has a directed temporal logic objective, we show how to efficiently search over the contract space and reduce the complexity of the original synthesis procedure.