This study presents a novel approach for active vibration control of a Shape Memory Alloy (SMA) von Mises structure using a bioinspired controller. A constitutive model that accurately captures the SMA thermomechanical behavior is used to describe the system dynamics. Nevertheless, this model is computationally demanding and requires the identification and adjustment of a large number of parameters, making its direct use in the controller design impractical for real-time control applications. In this context, the controller is based on the sliding mode control approach being enhanced with a neural network to compensate uncertainties and external disturbances, and it is designed using a polynomial constitutive model to approximate the system dynamics, as required by the control formulation. This model is adopted due to its simplicity and reduced number of parameters to be identified, while providing an approximate representation of the plant. Hence, the proposed control law is capable of predicting the system’s dynamical behavior, adapting to changes, and learning from the system–environment interaction in an online manner. The closed-loop system’s convergence is analyzed using a Lyapunov-like stability analysis, and numerical simulations demonstrate the effectiveness of the proposed strategy for suppressing undesired vibrations in the presence of nonlinear effects and model mismatch. Overall, the proposed approach combines reduced computational complexity with online adaptation capabilities, providing an efficient framework for active vibration control of SMA structures.