With the rapid development of web technology, more and more software applications have become web-based in the past decades. To ensure software quality and user experience, various techniques have been proposed to automatically test web applications by interacting with their GUIs. To achieve high functional coverage, web GUI testing tools often need to generate high-quality text inputs and interact with the associated GUI elements (e.g., click submit buttons). However, developing a holistic approach that solves both subtasks is challenging because the web GUI context can be complicated and highly dynamic, which makes it hard to process programmatically. The recent development of large vision-language models (LVLM) provides new opportunities to handle these longstanding problems. This paper proposes VETL, the first LVLM-driven end-to-end web testing technique. With LVLM's scene understanding capabilities, VETL can generate valid and meaningful text inputs focusing on the local context, while avoiding the need to extract precise textual attributes. The selection of associated GUI elements is formulated as a visual question-answering problem, allowing LVLM to capture the logical connection between the input box and the relevant element based on visual instructions. Further, the GUI exploration is guided by a multi-armed bandit module employing a curiosity-oriented strategy. Experiments show that VETL effectively explores web state/action spaces and detects bugs. Compared with WebExplor, the state-of-the-art web testing technique, VETL can discover 25% more unique web actions on benchmark websites. Moreover, it can expose functional bugs in top-ranking commercial websites, which the website maintainers have confirmed. Our work makes the first attempt at leveraging LVLM in end-to-end GUI testing, demonstrating promising results in this research direction.