Forecasting chaotic time series requires models that can capture the intrinsic geometry of the underlying attractor while remaining computationally efficient. We introduce a novel reservoir computing (RC) framework that integrates time-delay embedding with Random Fourier Feature (RFF) mappings to construct a dynamical reservoir without the need for traditional recurrent architectures. Unlike standard RC, which relies on high-dimensional recurrent connectivity, the proposed RFF-RC explicitly approximates nonlinear kernel transformations that uncover latent dynamical relations in the reconstructed phase space. This hybrid formulation offers two key advantages: (i) it provides a principled way to approximate complex nonlinear interactions among delayed coordinates, thereby enriching the effective dynamical representation of the reservoir, and (ii) it reduces reliance on manual reservoir hyperparameters such as spectral radius and leaking rate. We evaluate the framework on canonical chaotic systems-the Mackey-Glass equation, the Lorenz system, and the Kuramoto-Sivashinsky equation. This novel formulation demonstrates that RFF-RC not only achieves superior prediction accuracy but also yields robust attractor reconstructions and long-horizon forecasts. These results show that the combination of delay embedding and RFF-based reservoirs reveals new dynamical structure by embedding the system in an enriched feature space, providing a computationally efficient and interpretable approach to modeling chaotic dynamics.