Continuum robots, made from flexible materials with continuous backbones, have several advantages over traditional rigid robots. Some of them are the ability to navigate through narrow or confined spaces, adapt to irregular or changing environments, and perform tasks in proximity to humans. However, one of the challenges in using continuum robots is the difficulty in accurately estimating their state, such as their tip position and curvature. This is due to the complexity of their kinematics and the inherent uncertainty in their measurement and control. This paper proposes a moving horizon estimation (MHE) approach for estimating the robot's state, including its tip position and shape parameters. Our approach involves minimizing the error between measurement samples from an IMU attached to the robot's tip and the estimated state along the estimation horizon using an inline optimization problem. We demonstrate the effectiveness of our approach through simulation and experimental results. Our approach can potentially improve the accuracy and robustness of state estimation and control for continuum robots. It can be applied to various applications such as surgery, manufacturing, and inspection.