This paper presents the first flight demonstration using machine learning-based warm start to accelerate trajectory optimization on the free-flying Astrobee robot aboard the International Space Station (ISS). We present a data-driven optimal control method that trains a neural network that learns the structure of a trajectory generation problem solved using sequential convex programming (SCP). The trained neural network predicts a solution to the trajectory generation problem and enforces safety constraints on the system using an SCP solver. We demonstrate a 60% reduction in the number of solver iterations when rotational dynamics are included, and a 50% reduction when obstacles drawn from the training distribution of the warm start model are present. This represents a significant milestone in the use of learning-based control in spaceflight applications and serves as a stepping stone for future advances in machine learning for autonomous guidance, navigation, and control.