Contrary to the stunning feats observed in birds of prey, aerial manipulation and grasping with flying robots still lack versatility and agility. Conventional approaches using rigid manipulators require precise positioning and are subject to large reaction forces at grasp, which limit performance at high speeds. The few reported examples of aggressive aerial grasping rely on motion capture systems, or fail to generalize across environments and grasp targets. We describe the first example of a soft aerial manipulator equipped with a fully onboard perception pipeline, capable of robustly localizing and grasping visually and morphologically varied objects. The proposed system features a novel passively closing tendon-actuated soft gripper that enables fast closure at grasp, while compensating for position errors, complying to the target-object morphology, and dampening reaction forces. The system includes an onboard perception pipeline that combines a neural-network-based semantic keypoint detector with a state-of-the-art robust 3D object pose estimator, whose estimate is further refined using a fixed-lag smoother. The resulting pose estimate is passed to a minimum-snap trajectory planner, tracked by an adaptive controller that fully compensates for the added mass of the grasped object. Finally, a finite-element-based controller determines optimal gripper configurations for grasping. Rigorous experiments confirm that our approach enables dynamic, aggressive, and versatile grasping. We demonstrate fully onboard vision-based grasps of a variety of objects, in both indoor and outdoor environments, and up to speeds of 2.0 m/s -- the fastest vision-based grasp reported in the literature. Finally, we take a major step in expanding the utility of our platform beyond stationary targets, by demonstrating motion-capture-based grasps of targets moving up to 0.3 m/s, with relative speeds up to 1.5 m/s.