Vector search underpins modern AI applications by supporting approximate nearest neighbor (ANN) queries over high-dimensional embeddings in tasks like retrieval-augmented generation (RAG), recommendation systems, and multimodal search. Traditional ANN search indices (e.g., HNSW) are limited by memory constraints at large data scale. Disk-based indices such as DiskANN reduce memory overhead but rely on offline graph construction, resulting in costly and inefficient vector updates. The state-of-the-art clustering-based approach SPFresh offers better scalability but suffers from reduced recall due to coarse partitioning. Moreover, SPFresh employs in-place updates to maintain its index structure, limiting its efficiency in handling high-throughput insertions and deletions under dynamic workloads. This paper presents LSM-VEC, a disk-based dynamic vector index that integrates hierarchical graph indexing with LSM-tree storage. By distributing the proximity graph across multiple LSM-tree levels, LSM-VEC supports out-of-place vector updates. It enhances search efficiency via a sampling-based probabilistic search strategy with adaptive neighbor selection, and connectivity-aware graph reordering further reduces I/O without requiring global reconstruction. Experiments on billion-scale datasets demonstrate that LSM-VEC consistently outperforms existing disk-based ANN systems. It achieves higher recall, lower query and update latency, and reduces memory footprint by over 66.2%, making it well-suited for real-world large-scale vector search with dynamic updates.