The continuum robotic arm with a central backbone configuration utilizes a single continuous medium rod as its main structure, lacking the joint mechanisms found in traditional robotic arms. This design makes shape feedback a long-standing challenge. Currently, most shape measurement methods based on multiple IMUs rely on the piecewise constant curvature assumption. However, this assumption often fails when the robotic arm is subjected to external loads, leading to reduced accuracy in shape estimation. To address this issue, this paper proposes a multi-IMU shape estimation algorithm for continuum robotic arms based on tangent vector fitting. The algorithm employs Cosserat rod theory to mathematically model the continuum robotic arm, enabling a more accurate description of its deformation behavior. The shape at multiple measurement points is estimated using error-state Kalman filtering, and the tangent vector at each point is calculated. Subsequently, B-spline fitting is applied to the discrete tangent vectors to obtain a continuous tangent vector function with arc length as the independent variable. Finally, by integrating this continuously varying tangent vector function, the shape estimation is completed. Experimental results demonstrate that the algorithm achieves high-precision shape estimation under both dynamic trajectories and static loads, particularly when significant shape changes are induced by external loads. The algorithm exhibits strong robustness and stability. Compared to traditional methods based on the piecewise constant curvature assumption, the proposed algorithm significantly improves the positioning accuracy at the end-effector and the accuracy of shape reconstruction. Under high-load conditions, the shape estimation error is reduced by more than 50% compared to existing methods, proving its superiority in complex application scenarios.