Vertical Federated Learning (VFL) enables collaborative learning by integrating disjoint feature subsets from multiple clients/parties. However, VFL typically faces two key challenges: i) the requirement for perfectly aligned data samples across all clients (missing features are not allowed); ii) the requirement for joint collaborative inference/prediction involving all clients (it does not support locally independent inference on a single client). To address these challenges, we propose X-VFL, a new VFL framework designed to deal with the non-aligned data samples with (partially) missing features and to support locally independent inference of new data samples for each client. In particular, we design two novel modules in X-VFL: Cross Completion (XCom) and Decision Subspace Alignment (DS-Align). XCom can complete/reconstruct missing features for non-aligned data samples by leveraging information from other clients. DS-Align aligns local features with completed and global features across all clients within the decision subspace, thus enabling locally independent inference at each client. Moreover, we provide convergence theorems for different algorithms used in training X-VFL, showing an $O(1/\sqrt{T})$ convergence rate for SGD-type algorithms and an $O(1/T)$ rate for PAGE-type algorithms, where $T$ denotes the number of training update steps. Extensive experiments on real-world datasets demonstrate that X-VFL significantly outperforms existing methods, e.g., achieving a 15% improvement in accuracy on the image CIFAR-10 dataset and a 43% improvement on the medical MIMIC-III dataset. These results validate the practical effectiveness and superiority of X-VFL, particularly in scenarios involving partially missing features and locally independent inference.