Rare diseases affect an estimated 300-400 million people worldwide, yet individual conditions remain underdiagnosed and poorly characterized due to their low prevalence and limited clinician familiarity. Computational phenotyping offers a scalable approach to improving rare disease detection, but algorithm development is hindered by the scarcity of high-quality labeled data for training. Expert-labeled datasets from chart reviews and registries are clinically accurate but limited in scope and availability, whereas labels derived from electronic health records (EHRs) provide broader coverage but are often noisy or incomplete. To address these challenges, we propose WEST (WEakly Supervised Transformer for rare disease phenotyping and subphenotyping from EHRs), a framework that combines routinely collected EHR data with a limited set of expert-validated cases and controls to enable large-scale phenotyping. At its core, WEST employs a weakly supervised transformer model trained on extensive probabilistic silver-standard labels - derived from both structured and unstructured EHR features - that are iteratively refined during training to improve model calibration. We evaluate WEST on two rare pulmonary diseases using EHR data from Boston Children's Hospital and show that it outperforms existing methods in phenotype classification, identification of clinically meaningful subphenotypes, and prediction of disease progression. By reducing reliance on manual annotation, WEST enables data-efficient rare disease phenotyping that improves cohort definition, supports earlier and more accurate diagnosis, and accelerates data-driven discovery for the rare disease community.