Precision agriculture is rapidly attracting research to efficiently introduce automation and robotics solutions to support agricultural activities. Robotic navigation in vineyards and orchards offers competitive advantages in autonomously monitoring and easily accessing crops for harvesting, spraying and performing time-consuming necessary tasks. Nowadays, autonomous navigation algorithms exploit expensive sensors which also require heavy computational cost for data processing. Nonetheless, vineyard rows represent a challenging outdoor scenario where GPS and Visual Odometry techniques often struggle to provide reliable positioning information. In this work, we combine Edge AI with Deep Reinforcement Learning to propose a cutting-edge lightweight solution to tackle the problem of autonomous vineyard navigation without exploiting precise localization data and overcoming task-tailored algorithms with a flexible learning-based approach. We train an end-to-end sensorimotor agent which directly maps noisy depth images and position-agnostic robot state information to velocity commands and guides the robot to the end of a row, continuously adjusting its heading for a collision-free central trajectory. Our extensive experimentation in realistic simulated vineyards demonstrates the effectiveness of our solution and the generalization capabilities of our agent.