Few-shot learning (FSL) via customization of a deep learning network with limited data has emerged as a promising technique to achieve personalized user experiences on edge devices. However, existing FSL methods primarily assume independent and identically distributed (IID) data and utilize either computational backpropagation updates for each task or a common model with task-specific prototypes. Unfortunately, the former solution is infeasible for edge devices that lack on-device backpropagation capabilities, while the latter often struggles with limited generalization ability, especially for out-of-distribution (OOD) data. This paper proposes a lightweight, plug-and-play FSL module called Task-aware Normalization (TANO) that enables efficient and task-aware adaptation of a deep neural network without backpropagation. TANO covers the properties of multiple user groups by coordinating the updates of several groups of the normalization statistics during meta-training and automatically identifies the appropriate normalization group for a downstream few-shot task. Consequently, TANO provides stable but task-specific estimations of the normalization statistics to close the distribution gaps and achieve efficient model adaptation. Results on both intra-domain and out-of-domain generalization experiments demonstrate that TANO outperforms recent methods in terms of accuracy, inference speed, and model size. Moreover, TANO achieves promising results on widely-used FSL benchmarks and data from real applications.