Underwater caves are challenging environments that are crucial for water resource management, and for our understanding of hydro-geology and history. Mapping underwater caves is a time-consuming, labor-intensive, and hazardous operation. For autonomous cave mapping by underwater robots, the major challenge lies in vision-based estimation in the complete absence of ambient light, which results in constantly moving shadows due to the motion of the camera-light setup. Thus, detecting and following the caveline as navigation guidance is paramount for robots in autonomous cave mapping missions. In this paper, we present a computationally light caveline detection model based on a novel Vision Transformer (ViT)-based learning pipeline. We address the problem of scarce annotated training data by a weakly supervised formulation where the learning is reinforced through a series of noisy predictions from intermediate sub-optimal models. We validate the utility and effectiveness of such weak supervision for caveline detection and tracking in three different cave locations: USA, Mexico, and Spain. Experimental results demonstrate that our proposed model, CL-ViT, balances the robustness-efficiency trade-off, ensuring good generalization performance while offering 10+ FPS on single-board (Jetson TX2) devices.