The increasing complexity of space missions has highlighted the critical challenge of vibration suppression in ultra-large space structures such as solar arrays, trusses, and antennas. These structures are prone to low-frequency vibrations during orbital maneuvers, significantly degrading spacecraft attitude stability and positioning accuracy. While traditional state-estimation-based control methods suffer from high model complexity and noise sensitivity, policy gradient methods that directly optimize control gains face limitations in convergence speed and hyperparameter tuning. To address these issues, this paper proposes a novel output-feedback Linear Quadratic Regulator (LQR) control framework integrating Bayesian Optimization (BO) with policy gradient projection. Our approach formulates the gain search process as a black-box optimization problem, employing Gaussian Processes (GP) to construct surrogate models and acquisition functions to balance exploration-exploitation trade-offs while satisfying system performance constraints. Key contributions include: (1) embedding BO within the policy update loop to accelerate convergence and enhance data efficiency, and (2) demonstrating superior performance through numerical simulations on piezoelectric cantilever beam systems compared to existing methods. Experimental results confirm the framework's effectiveness for vibration control in complex space structures, with additional discussions on potential extensions to adaptive control and multi-agent coordination scenarios.