In this paper, we introduce a relatively simple and versatile analytical model to compute the performance of an IEEE 802.11-based Video-on-Demand (VoD) service for connected vehicles traveling along a highway or an expressway. We use a probabilistic approach to account for the intrinsic dynamics (mobility) of vehicles, the density of subscriber vehicles, the distance between Road Side Units (RSUs), as well as the video codec bit rate. Our solution delivers estimates for the attained throughput of each subscriber vehicle as well as for their interruption times in the playback of their video. We studied several scenarios to assess the accuracy of our approximate solution comparing its results with those of a discrete-event simulator with realistic vehicular traffic trajectories, video traffic, and networking protocols. Overall, our solution is found to be accurate delivering estimates in good agreement with the simulation results. Through several applications, we illustrate how our analytical solution can help in the design of a VoD service for vehicles for capacity planning purposes, or at improving its performance with the use of a blocking policy.