Pixel-wise annotations are notoriously labourious and costly to obtain in the medical domain. To mitigate this burden, weakly supervised approaches based on bounding box annotations-much easier to acquire-offer a practical alternative. Vision foundation models have recently shown noteworthy segmentation performance when provided with prompts such as points or bounding boxes. Prompt learning exploits these models by adapting them to downstream tasks and automating segmentation, thereby reducing user intervention. However, existing prompt learning approaches depend on fully annotated segmentation masks. This paper proposes a novel framework that combines the representational power of foundation models with the annotation efficiency of weakly supervised segmentation. More specifically, our approach automates prompt generation for foundation models using only bounding box annotations. Our proposed optimization scheme integrates multiple constraints derived from box annotations with pseudo-labels generated by the prompted foundation model. Extensive experiments across multimodal datasets reveal that our weakly supervised method achieves an average Dice score of 84.90% in a limited data setting, outperforming existing fully-supervised and weakly-supervised approaches. The code is available at https://github.com/Minimel/box-prompt-learning-VFM.git