In this paper, we study the control design of an automatic crosswind stabilization system for a novel, buoyantly-assisted aerial transportation vehicle. This vehicle has several advantages over other aircraft including the ability to take-off and land in very short distances and without the need for roads or runways. Despite these advantages, the large surface area of the vehicle's wing makes it more susceptible to wind, which introduces undesirable roll angle motions. The role of the automatic crosswind stabilization system is to detect the roll angle deviation, and then use motors at the wingtips to counteract the wind effect. However, due to the relatively large inertia of the wing compared to small-size unmanned aerial vehicles and additional input time delays, an automatic crosswind stabilization system based on traditional control algorithms such as the proportional-integral-derivative (PID) controller results in a response time that is too slow. Another challenge is the lack of high-accuracy wind sensors that can be mounted on the vehicle's wing. Therefore, we first design a wind torque estimator that relies on inertial measurements, and then use feed-forward compensation to directly correct for the wind torque, resulting in a significantly faster response. We second combine the proposed estimator with a model predictive controller (MPC), and compare constrained MPC with unconstrained MPC for the considered application. Experimental results show that our proposed estimation-based MPC strategy reduces the response time of the system by around 80-90% compared to a standard PID controller, without the need for adding wind sensors or changing the hardware of the stabilization system.