This paper considers a radio-frequency (RF)-based simultaneous localization and source-seeking (SLASS) problem in multi-robot systems, where multiple robots jointly localize themselves and an RF source using distance-only measurements extracted from RF signals and then control themselves to approach the source. We design a Rao-Blackwellized particle filter-based algorithm to realize the joint localization of the robots and the source. We also devise an information-theoretic control policy for the robots to approach the source. In our control policy, we maximize the predicted mutual information between the source position and the distance measurements, conditioned on the robot positions, to incorporate the robot localization uncertainties. A projected gradient ascent method is adopted to solve the mutual information maximization problem. Simulation results show that the proposed SLASS framework outperforms two benchmarks in terms of the root mean square error (RMSE) of the estimated source position and the decline of the distances between the robots and the source, indicating more effective approaching of the robots to the source.