While transformers demonstrate outstanding performance across various audio tasks, their application to neural vocoders remains challenging. Neural vocoders require the generation of long audio signals at the sample level, which demands high temporal resolution. This results in significant computational costs for attention map generation and limits their ability to efficiently process both global and local information. Additionally, the sequential nature of sample generation in neural vocoders poses difficulties for real-time processing, making the direct adoption of transformers impractical. To address these challenges, we propose RingFormer, a neural vocoder that incorporates the ring attention mechanism into a lightweight transformer variant, the convolution-augmented transformer (Conformer). Ring attention effectively captures local details while integrating global information, making it well-suited for processing long sequences and enabling real-time audio generation. RingFormer is trained using adversarial training with two discriminators. The proposed model is applied to the decoder of the text-to-speech model VITS and compared with state-of-the-art vocoders such as HiFi-GAN, iSTFT-Net, and BigVGAN under identical conditions using various objective and subjective metrics. Experimental results show that RingFormer achieves comparable or superior performance to existing models, particularly excelling in real-time audio generation. Our code and audio samples are available on GitHub.