A novel approach for linear parameter-varying (LPV) controller synthesis for adaptive rejection of time-varying sinusoidal disturbances is proposed. Only the frequency response data of a linear time-invariant (LTI) multiple-input multiple-output (MIMO) system is used to design the LPV controller that stabilizes the system for arbitrary fast variation of the scheduling parameter. Global stability is achieved thanks to the specific structure of the LPV controller and the use of integral quadratic constraints (IQC) to represent the variation of the scheduling parameter. The LPV controller is designed by convex optimization in the frequency domain. For experimental validation of the proposed method, a hybrid micro-vibration damping platform for space applications is considered. An LPV controller for rejection of unknown time-varying disturbances is designed and implemented on the real system. Experimental results show the effectiveness of the proposed approach for rejection of disturbances and closed-loop stability for arbitrarily fast time-varying scheduling signals.