Multimodal deep learning has shown strong potential in medical applications by integrating heterogeneous data sources such as medical images and structured clinical variables. However, most existing approaches implicitly assume complete modality availability, an assumption that rarely holds in real-world clinical settings where entire modalities and individual features are frequently missing. In this work, we propose a multimodal transformer framework for joint vision-tabular learning explicitly designed to operate under pervasive modality missingness, without relying on imputation or heuristic model switching. The architecture integrates three components: a vision, a tabular, and a multimodal fusion encoder. Unimodal representations are weighted through learnable modality tokens and fused via intermediate fusion with masked self-attention, which excludes missing tokens and modalities from information aggregation and gradient propagation. To further enhance resilience, we introduce a modality-dropout regularization strategy that stochastically removes available modalities during training, encouraging the model to exploit complementary information under partial data availability. We evaluate our approach on the MIMIC-CXR dataset paired with structured clinical data from MIMIC-IV for multilabel classification of 14 diagnostic findings with incomplete annotations. Two parallel systematic stress-test protocols progressively increase training and inference missingness in each modality separately, spanning fully multimodal to fully unimodal scenarios. Across all missingness regimes, the proposed method consistently outperforms representative baselines, showing smoother performance degradation and improved robustness. Ablation studies further demonstrate that attention-level masking and intermediate fusion with joint fine-tuning are key to resilient multimodal inference.