Solving Inverse Kinematics (IK) for arbitrary kinematic trees presents significant challenges due to their high-dimensionality, redundancy, and complex inter-branch constraints. Conventional optimization-based solvers can be sensitive to initialization and suffer from local minima or conflicting gradients. At the same time, existing learning-based approaches are often tied to a predefined number of end-effectors and a fixed training objective, limiting their reusability across various robot morphologies and task requirements. To address these limitations, we introduce IKDiffuser, a scalable IK solver built upon conditional diffusion-based generative models, which learns the distribution of the configuration space conditioned on end-effector poses. We propose a structure-agnostic formulation that represents end-effector poses as a sequence of tokens, leading to a unified framework that handles varying numbers of end-effectors while learning the implicit kinematic structures entirely from data. Beyond standard IK generation, IKDiffuser handles partially specified goals via a masked marginalization mechanism that conditions only on a subset of end-effector constraints. Furthermore, it supports adding task objectives at inference through objective-guided sampling, enabling capabilities such as warm-start initialization and manipulability maximization without retraining. Extensive evaluations across seven diverse robotic platforms demonstrate that IKDiffuser significantly outperforms state-of-the-art baselines in accuracy, solution diversity, and collision avoidance. Moreover, when used to initialize optimization-based solvers, IKDiffuser significantly boosts success rates on challenging redundant systems with high Degrees of Freedom (DoF), such as the 29-DoF Unitree G1 humanoid, from 21.01% to 96.96% while reducing computation time to the millisecond range.