This paper introduces the consistent vehicle routing problem with stochastic customers and demands. We consider driver consistency as customer-driver assignments that remain fixed when the realizations of the random variables are observed. We study the problem in a two-stage scenario-based stochastic programming framework. In the first stage, customers are assigned to drivers, while in the second stage, customers are selected and delivery routes are designed for each of the scenarios. We assume that the realization of the random variables becomes known before the vehicles depart from the depot. The routes are then optimized according to the observed customers and their demands. The first-stage driver-customer assignments can violate the consistency requirement, which is modeled as a desired maximum number of drivers assigned to each customer. This is modeled as a soft constraint with a penalty in the objective function. It is hence possible to assign multiple drivers to a specific customer in the first stage. In the second stage, a customer can only be visited by one of the preassigned drivers. Our problem, therefore, consists in finding assignments that minimize the consistency violation penalties and the expected routing costs and the penalties for unserved customers when the uncertain parameters are revealed. We present a mathematical formulation and a sample average approximation~(SAA) approach for the problem. We introduce a branch-and-cut and a Benders decomposition method to solve the sample problems in our SAA algorithm. Computational experiments show that~SAA allows finding good-quality solutions for instances with large sets of scenarios. We also analyze the cost-consistency trade-offs and the impact of the uncertainty on the problem. In particular, we observe that consistency can be promoted through a flexible approach that does not compromise excessively on other operational metrics. Furthermore, we analyze the impact of not considering the problem uncertainties during the planning stage.