Vertical federated learning trains models from feature-partitioned datasets across multiple clients, who collaborate without sharing their local data. Standard approaches assume that all feature partitions are available during both training and inference. Yet, in practice, this assumption rarely holds, as for many samples only a subset of the clients observe their partition. However, not utilizing incomplete samples during training harms generalization, and not supporting them during inference limits the utility of the model. Moreover, if any client leaves the federation after training, its partition becomes unavailable, rendering the learned model unusable. Missing feature blocks are therefore a key challenge limiting the applicability of vertical federated learning in real-world scenarios. To address this, we propose LASER-VFL, a vertical federated learning method for efficient training and inference of split neural network-based models that is capable of handling arbitrary sets of partitions. Our approach is simple yet effective, relying on the sharing of model parameters and on task-sampling to train a family of predictors. We show that LASER-VFL achieves a $\mathcal{O}({1}/{\sqrt{T}})$ convergence rate for nonconvex objectives and, under the Polyak-Łojasiewicz inequality, it achieves linear convergence to a neighborhood of the optimum. Numerical experiments show improved performance of LASER-VFL over the baselines. Remarkably, this is the case even in the absence of missing features. For example, for CIFAR-100, we see an improvement in accuracy of $19.3$\% when each of four feature blocks is observed with a probability of 0.5 and of $9.5$\% when all features are observed. The code for this work is available at https://github.com/Valdeira/LASER-VFL.