Deep learning has brought about a revolution in 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, and manual design can be prohibitively expensive for customized NNs tailored to different scenarios. To tackle this challenge, this paper proposes the use of neural architecture search (NAS) to automate the generation of scenario-customized CSI feedback NN architectures. By employing automated machine learning and gradient-descent-based NAS, an efficient and cost-effective architecture design process is achieved, requiring less expert experience and design time, thus lowering the design threshold. The proposed approach leverages implicit scene knowledge and integrates it into the scenario customization process in a data-driven manner, fully exploiting the potential of deep learning in a given scenario. To address the issue of excessive search, early stopping and elastic selection mechanisms are employed, further enhancing the proposed scheme. The experimental results demonstrate that the generated architecture, known as Auto-CsiNet, outperforms manually-designed models in terms of reconstruction performance (achieving approximately 14% improvement) and complexity (reducing by approximately 50%), highlighting the effectiveness of the NAS-based automatic scheme. Furthermore, the paper analyzes the impact of the scenario on the NN architecture and capacity.