In recent years, high-speed navigation and environment interaction in the context of aerial robotics has become a field of interest for several academic and industrial research studies. In particular, Search and Intercept (SaI) applications for aerial robots pose a compelling research area due to their potential usability in several environments. Nevertheless, SaI tasks involve a challenging development regarding sensory weight, on-board computation resources, actuation design and algorithms for perception and control, among others. In this work, a fully-autonomous aerial robot for high-speed object grasping has been proposed. As an additional sub-task, our system is able to autonomously pierce balloons located in poles close to the surface. Our first contribution is the design of the aerial robot at an actuation and sensory level consisting of a novel gripper design with additional sensors enabling the robot to grasp objects at high speeds. The second contribution is a complete software framework consisting of perception, state estimation, motion planning, motion control and mission control in order to rapid- and robustly perform the autonomous grasping mission. Our approach has been validated in a challenging international competition and has shown outstanding results, being able to autonomously search, follow and grasp a moving object at 6 m/s in an outdoor environment