Recent advances in self-supervision and constrastive learning have brought the performance of foundation models to unprecedented levels in a variety of tasks. Fueled by this progress, these models are becoming the prevailing approach for a wide array of real-world vision problems, including risk-sensitive and high-stakes applications. However, ensuring safe deployment in these scenarios requires a more comprehensive understanding of their uncertainty modeling capabilities, which has been barely explored. In this work, we delve into the behavior of vision and vision-language foundation models under Conformal Prediction (CP), a statistical framework that provides theoretical guarantees of marginal coverage of the true class. Across extensive experiments including popular vision classification benchmarks, well-known foundation vision models, and three CP methods, our findings reveal that foundation models are well-suited for conformalization procedures, particularly those integrating Vision Transformers. Furthermore, we show that calibrating the confidence predictions of these models leads to efficiency degradation of the conformal set on adaptive CP methods. In contrast, few-shot adaptation to downstream tasks generally enhances conformal scores, where we identify Adapters as a better conformable alternative compared to Prompt Learning strategies. Our empirical study identifies APS as particularly promising in the context of vision foundation models, as it does not violate the marginal coverage property across multiple challenging, yet realistic scenarios.