Operator learning is the approximation of operators between infinite dimensional Banach spaces using machine learning approaches. While most progress in this area has been driven by variants of deep neural networks such as the Deep Operator Network and Fourier Neural Operator, the theoretical guarantees are often in the form of a universal approximation property. However, the existence theorems do not guarantee that an accurate operator network is obtainable in practice. Motivated by the recent kernel-based operator learning framework, we propose a random feature operator learning method with theoretical guarantees and error bounds. The random feature method can be viewed as a randomized approximation of a kernel method, which significantly reduces the computation requirements for training. We provide a generalization error analysis for our proposed random feature operator learning method along with comprehensive numerical results. Compared to kernel-based method and neural network methods, the proposed method can obtain similar or better test errors across benchmarks examples with significantly reduced training times. An additional advantages it that our implementation is simple and does require costly computational resources, such as GPU.