In this paper, the non-orthogonal multiple access (NOMA) technology is applied in a vehicular edge computing network, in which mobile vehicles can offload partial computation tasks to the MEC server for remote execution. In this network, a long-term energy consumption minimization problem is presented by jointly optimizing the successive interference cancellation (SIC) order, transmit power, and computation resource allocation. To deal with the formulated problem, we first transform it into an equivalent instantaneous form based on the Lyapunov optimization theory. Since the transformed problem is still highly non-convex, we further decompose it into the interactive resource allocation and SIC order subproblems. For the resource allocation subproblem, we exploit its convexity through the transformation and reparameterization and then derive the optimal solution by the Karush-Kuhn-Tucker (KKT) conditions. After that, we propose a low-complexity algorithm by leveraging tabu search to obtain the suboptimal SIC order. Simulation results validate the effectiveness of the proposed algorithm and the superiority of NOMA compared to frequency division multiple access (FDMA).