Large language models (LLMs) exhibit diverse response behaviors, costs, and strengths, making it challenging to select the most suitable LLM for a given user query. We study the problem of adaptive multi-LLM selection in an online setting, where the learner interacts with users through multi-step query refinement and must choose LLMs sequentially without access to offline datasets or model internals. A key challenge arises from unstructured context evolution: the prompt dynamically changes in response to previous model outputs via a black-box process, which cannot be simulated, modeled, or learned. To address this, we propose the first contextual bandit framework for sequential LLM selection under unstructured prompt dynamics. We formalize a notion of myopic regret and develop a LinUCB-based algorithm that provably achieves sublinear regret without relying on future context prediction. We further introduce budget-aware and positionally-aware (favoring early-stage satisfaction) extensions to accommodate variable query costs and user preferences for early high-quality responses. Our algorithms are theoretically grounded and require no offline fine-tuning or dataset-specific training. Experiments on diverse benchmarks demonstrate that our methods outperform existing LLM routing strategies in both accuracy and cost-efficiency, validating the power of contextual bandits for real-time, adaptive LLM selection.