State estimation in the presence of unknown inputs presents a significant challenge in many domains. Numerous works in the literature have addressed this issue within linear systems, and only a few explored the challenges of nonlinear cases. Recognizing the importance of state estimation on the special orthogonal group SO(3), particularly in attitude estimation, this paper presents a state estimation algorithm designed for SO(3) with unknown inputs, specifically for cases where the unknown input affects the state dynamics without direct feedthrough to the output. Furthermore, this paper demonstrates an application in gyro-free attitude estimation. Monte Carlo simulations show better accuracy and performance over existing attitude estimation methods using only a 3-axis accelerometer and a 3-axis magnetometer.