In this study, we propose VibeCodeHPC, a multi-agent system based on large language models (LLMs) for the automatic tuning of high-performance computing (HPC) programs on supercomputers. VibeCodeHPC adopts Claude Code as its backend and provides an integrated environment that facilitates program development in supercomputer settings. The system not only brings the Vibe Coding paradigm -- program development through natural language interaction with users -- to HPC programming, but also enables autonomous performance optimization with minimal user intervention through a sophisticated multi-agent design. To achieve these objectives, VibeCodeHPC implements three core functionalities: (1) configuration capabilities tailored to the unique development environments of supercomputers, (2) collaborative operation among multiple LLM agents with distinct roles -- Project Manager (PM), System Engineer (SE), Programmer (PG), and Continuous Deliverer (CD), and (3) long-term autonomous operation through agent activity monitoring and dynamic deployment mechanisms. This paper highlights one of the most powerful features of VibeCodeHPC: fully automated code optimization through autonomous operation without user intervention. Specifically, it demonstrates the performance optimization of CPU-based codes on GPU-equipped systems for matrix multiplication and a Poisson equation solver using Jacobi's iterative method. The results show that the multi-agent configuration employed in VibeCodeHPC enables faster and more reliable development of higher-performance code compared to a single-agent setup.