Deep learning has revolutionized the design of the channel state information (CSI) feedback module in wireless communications. However, designing the optimal neural network (NN) architecture for CSI feedback can be a laborious and time-consuming process. Manual design can be prohibitively expensive for customizing NNs to different scenarios. This paper proposes using neural architecture search (NAS) to automate the generation of scenario-customized CSI feedback NN architectures, thereby maximizing the potential of deep learning in exclusive environments. By employing automated machine learning and gradient-descent-based NAS, an efficient and cost-effective architecture design process is achieved. The proposed approach leverages implicit scene knowledge, integrating it into the scenario customization process in a data-driven manner, and fully exploits the potential of deep learning for each specific scenario. To address the issue of excessive search, early stopping and elastic selection mechanisms are employed, enhancing the efficiency of the proposed scheme. The experimental results demonstrate that the automatically generated architecture, known as Auto-CsiNet, outperforms manually-designed models in both reconstruction performance (achieving approximately a 14% improvement) and complexity (reducing it by approximately 50%). Furthermore, the paper analyzes the impact of the scenario on the NN architecture and its capacity.