With the rapid growth of emerging delay-sensitive and computation-intensive vehicular applications, the integration of vehicle-to-everything (V2X) networks and multi-access edge computing (MEC) has emerged as a promising paradigm. However, how to make appropriate offloading decisions to reduce the overall processing time has become a critical issue. In this paper, we propose a novel V2X collaborative MEC network (VCMN) architecture that enables inter-roadside unit (RSU) collaboration for distributed task offloading. Based on this architecture, we formulate the task offloading problem as a nonlinear programming optimization problem, aiming to minimize the average total processing time of all offloading tasks while satisfying the processing time constraint of each offloading tasks. We propose an event-driven deep reinforcement learning-based task offloading policy (EDRL-TOP), which utilizes a centralized RSU manager to monitor the load conditions of all RSUs and RSU-to-RSU links, and intelligently makes offloading decisions for each event. The proposed EDRL-TOP fully utilizes a DRL technique, deep deterministic policy gradient (DDPG), to effectively handle the stochastic request arrivals thereby achieving long-term network performance optimization. Simulation results show that the proposed EDRL-TOP can decrease the average total processing time of tasks and achieves a higher task success rate compared to the greedy algorithm.