Self-supervised online traversability estimation enables robots to continuously learn from unlabeled open-world experiences and adapt their navigation behavior toward safe and efficient trajectories. Existing approaches either rely on handcrafted proprioceptive traversability scores, limiting robot-agnosticism, or cluster prior data, preventing online learning. Moreover, many continual learning methods incur substantial memory and computational costs, hindering onboard deployment. We introduce COTRATE, an online learning framework for continuous traversability estimation from multimodal, unlabeled robot experience. Our method first infers robust traversability scores using a robot-agnostic, learning-based online terrain assessment module operating on proprioceptiveand inertial signals. These scores then supervise a visual traversability network through a novel alignment loss that associates visual embeddings with online terrain assessments. To mitigate forgetting during continual learning with minimal overhead, we propose a diversity-aware feature selection strategythat preserves performance using a compact replay memory. We further show that the learned traversability representation supports knowledge transfer across different robot platforms with different locomotion kinematics. We evaluate COTRATE on a dataset of $\approx$ 50,000 images collected with two robotic platforms across 11 outdoor terrains, and benchmark it on navigation tasks in three representative outdoor environments. We make the dataset, code, and trained models publicly available.