In this paper, we model a transportation network by a Cox process where the road systems are modeled by a Manhattan Poisson line process (MPLP) and the locations of vehicles and desired destination sites, such as gas stations or charging stations, referred to as facilities, are modeled by independent 1D Poisson point processes (PPP) on each of the lines. For this setup, we characterize the length of the shortest path between a typical vehicular user and its nearest facility that can be reached by traveling along the streets. For a typical vehicular user starting from an intersection, we derive the closed-form expression for the exact cumulative distribution function (CDF) of the length of the shortest path to its nearest facility in the sense of path distance. Building on this result, we derive an upper bound and a remarkably accurate but approximate lower bound on the CDF of the shortest path distance to the nearest facility for a typical vehicle starting from an arbitrary position on a road. These results can be interpreted as nearest-neighbor distance distributions (in terms of the path distance) for this Cox process, which is a key technical contribution of this paper. In addition to these analytical results, we also present a simulation procedure to characterize any distance-dependent cost metric between a typical vehicular user and its nearest facility in the sense of path distance using graphical interpretation of the spatial model. We also discuss extension of this work to other cost metrics and possible applications to the areas of urban planning, personnel deployment and wireless communication.