Clustering clients into groups that exhibit relatively homogeneous data distributions represents one of the major means of improving the performance of federated learning (FL) in non-independent and identically distributed (non-IID) data settings. Yet, the applicability of current state-of-the-art approaches remains limited as these approaches cluster clients based on information, such as the evolution of local model parameters, that is only obtainable through actual on-client training. On the other hand, there is a need to make FL models available to clients who are not able to perform the training themselves, as they do not have the processing capabilities required for training, or simply want to use the model without participating in the training. Furthermore, the existing alternative approaches that avert the training still require that individual clients have a sufficient amount of labeled data upon which the clustering is based, essentially assuming that each client is a data annotator. In this paper, we present REPA, an approach to client clustering in non-IID FL settings that requires neither training nor labeled data collection. REPA uses a novel supervised autoencoder-based method to create embeddings that profile a client's underlying data-generating processes without exposing the data to the server and without requiring local training. Our experimental analysis over three different datasets demonstrates that REPA delivers state-of-the-art model performance while expanding the applicability of cluster-based FL to previously uncovered use cases.