We consider the problem of dynamic vehicle routing under exact-time constraints on servicing demands. Demands are sequentially generated in an environment, and every demand needs to be serviced exactly after a fixed finite interval of time after it is generated. We design routing policies for a service vehicle to maximize the fraction of demands serviced at steady state. The main contributions are as follows. First, we demonstrate that this problem is described by an appropriate directed acyclic graph structure which leads to a computationally efficient routing algorithm based on a longest-path computation. Second, under the assumption of the demands being generated uniformly randomly in the environment and via a Poisson process in time, we provide two analytic lower bounds on the service fraction of the longest path policy. The first bound is relative to an optimal noncausal version of the policy, i.e., a policy based on knowledge of all future demand requests. The second bound is an explicit function of the vehicle dynamics and demand generation rate and, therefore, useful as a design tool. Finally, we present numerical results to support the analytic bounds.