This paper introduces the design, modeling, manufacturing, and testing of a Gripping Aerial Topology Optimized Robot (GATOR). The airframe of this unmanned aerial vehicle (UAV) is designed to be lightweight, structurally stiff, modular, and multi-functional. A Level-Set Topology Optimization (LSTO) method defines the external geometry of the frame, while the frame infill is controlled using a variable thickness latticing technique based on Finite Element Analysis (FEA) results. The UAV incorporates a soft robotic gripper, allowing the vehicle to collect delicate samples from the environment and perch for low-power use for extended periods. The bioinspired design and fabrication of a mountable soft robotic gripper are presented and the associated kinematics are derived for controls. To further decrease the weight of the designs a novel volume-changing material was introduced following careful characterization through Scanning Electron Microscopy (SEM) and tensile testing. The resulting platform leverages additive manufacturing using material extrusion technology and can be swiftly instrumented with propulsion and flight control systems. The presented modular design methodology can be applied to the rapid prototyping of a broad range of aerial platforms and lightweight structures.