Establishing and maintaining 5G mmWave vehicular connectivity poses a significant challenge due to high user mobility that necessitates frequent triggering of beam switching procedures. Departing from reactive beam switching based on the user device channel state feedback, proactive beam switching prepares in advance for upcoming beam switching decisions by exploiting accurate channel state information (CSI) prediction. In this paper, we develop a framework for autonomous self-trained CSI prediction for mmWave vehicular users where a base station (gNB) collects and labels a dataset that it uses for training recurrent neural network (RNN)-based CSI prediction model. The proposed framework exploits the CSI feedback from vehicular users combined with overhearing the C-V2X cooperative awareness messages (CAMs) they broadcast. We implement and evaluate the proposed framework using deepMIMO dataset generation environment and demonstrate its capability to provide accurate CSI prediction for 5G mmWave vehicular users. CSI prediction model is trained and its capability to provide accurate CSI predictions from various input features are investigated.