The physical-layer security (PLS) of wireless networks has witnessed significant attention in next-generation communication systems due to its potential toward enabling protection at the signal level in dense network environments. The growing trends toward smart mobility via sensor-enabled vehicles are transforming today's traffic environment into Internet of Vehicles (IoVs). Enabling PLS for IoVs would be a significant development considering the dense vehicular network environment in the near future. In this context, this article presents a PLS framework for a vehicular network consisting a legitimate receiver and an eavesdropper, both under the effect of interfering vehicles. The double-Rayleigh fading channel is used to capture the effect of mobility within the communication channel. The performance is analyzed in terms of the average secrecy capacity (ASC) and secrecy outage probability (SOP). We present the standard expressions for the ASC and SOP in alternative forms, to facilitate analysis in terms of the respective moment generating function (MGF) and characteristic function of the joint fading and interferer statistics. Closed-form expressions for the MGFs and characteristic functions were obtained and Monte Carlo simulations were provided to validate the results. Approximate expressions for the ASC and SOP were also provided, for easier analysis and insight into the effect of the network parameters. The results attest that the performance of the considered system was affected by the number of interfering vehicles as well as their distances. It was also demonstrated that the system performance closely correlates with the uncertainty in the eavesdropper's vehicle location.