The increasing demand for rich multimedia services and the emergence of the Internet-of-Things (IoT) pose challenging requirements for the next generation vehicular networks. Such challenges are largely related to high spectral efficiency and low latency requirements in the context of massive content delivery and increased connectivity. In this respect, caching and non-orthogonal multiple access (NOMA) paradigms have been recently proposed as potential solutions to effectively address some of these key challenges. In the present contribution, we introduce cache-aided NOMA as an enabling technology for vehicular networks. In this context, we first consider the full file caching case, where each vehicle caches and requests entire files using the NOMA principle. Without loss of generality, we consider a two-user vehicular network communication scenario under double Nakagami$-m$ fading conditions and propose an optimum power allocation policy. To this end, an optimization problem that maximizes the overall probability of successful decoding of files at each vehicle is formulated and solved. Furthermore, we consider the case of split file caching, where each file is divided into two parts. A joint power allocation optimization problem is formulated, where power allocation across vehicles and cached split files is investigated. The offered analytic results are corroborated by extensive results from computer simulations and interesting insights are developed. Indicatively, it is shown that the proposed caching-aided NOMA outperforms the conventional NOMA technique.