Uncertainty-aware prediction is essential for safe motion planning, especially when using learned models to forecast the behavior of surrounding agents. Conformal prediction is a statistical tool often used to produce uncertainty-aware prediction regions for machine learning models. Most existing frameworks utilizing conformal prediction-based uncertainty predictions assume that the surrounding agents are non-interactive. This is because in closed-loop, as uncertainty-aware agents change their behavior to account for prediction uncertainty, the surrounding agents respond to this change, leading to a distribution shift which we call endogenous distribution shift. To address this challenge, we introduce an iterative conformal prediction framework that systematically adapts the uncertainty-aware ego-agent controller to the endogenous distribution shift. The proposed method provides probabilistic safety guarantees while adapting to the evolving behavior of reactive, non-ego agents. We establish a model for the endogenous distribution shift and provide the conditions for the iterative conformal prediction pipeline to converge under such a distribution shift. We validate our framework in simulation for 2- and 3- agent interaction scenarios, demonstrating collision avoidance without resulting in overly conservative behavior and an overall improvement in success rates of up to 9.6% compared to other conformal prediction-based baselines.