We consider an optimal route selection problem for ride-hailing service providers in stochastic and nonlinear road traffic environments described by the Lighthill-Whitham-Richards model. We show that the considered class of optimal control problem has a structure to which Kappen's path-integral control formalism is applicable. The effectiveness of the control synthesis based on the forward-in-time Monte-Carlo sampling suggested by the path-integral control is investigated via numerical studies. The path-integral control framework is generalized to compute a team solution to a two-player route selection problem where two ride-hailing companies collaborate on a shared transportation infrastructure.