We present a fast trajectory optimization algorithm for the soft capture of uncooperative tumbling space objects. Our algorithm generates safe, dynamically feasible, and minimum-fuel trajectories for a six-degree-of-freedom servicing spacecraft to achieve soft capture (near-zero relative velocity at contact) between predefined locations on the servicer spacecraft and target body. We solve a convex problem by enforcing a convex relaxation of the field-of-view constraint, followed by a sequential convex program correcting the trajectory for collision avoidance. The optimization problems can be solved with a standard second-order cone programming solver, making the algorithm both fast and practical for implementation in flight software. We demonstrate the performance and robustness of our algorithm in simulation over a range of object tumble rates up to 10°/s.