Rapid advancements in 5G communication and the Internet of Things have prompted the development of cognitive radio sensing for spectrum monitoring and malicious attack detection. An end-to-end radio classification system is essential to realize efficient real-time monitoring at the edge. This work presents an end-to-end neuromorphic system enabled by an efficient spiking neural network (SNN) for radio classification. A novel hardware-efficient spiking encoding method is proposed leveraging the Sigma-Delta modulation mechanism in analog-to-digital converters. It requires no additional hardware components, simplifies the system design, and helps reduce conversion latency. Following a designed hardware-emulating conversion process, the classification performance is verified on two benchmark radio modulation datasets. A comparable accuracy to an artificial neural network (ANN) baseline with a difference of 0.30% is achieved on the dataset RADIOML 2018 with more realistic conditions. Further analysis reveals that the proposed method requires less power-intensive computational operations, leading to 22× lower computational energy consumption. Additionally, this method exhibits more than 99% accuracy on the dataset when the signal-to-noise ratio is above zero dB. The SNN-based classification module is realized on FPGA with a heterogeneous streaming architecture, achieving high throughput and low resource utilization. Therefore, this work demonstrates a promising solution for constructing an efficient high-performance end-to-end radio classification system.