The remarkable capabilities and intricate nature of Artificial Intelligence (AI) have dramatically escalated the imperative for specialized AI accelerators. Nonetheless, designing these accelerators for various AI workloads remains both labor- and time-intensive. While existing design exploration and automation tools can partially alleviate the need for extensive human involvement, they still demand substantial hardware expertise, posing a barrier to non-experts and stifling AI accelerator development. Motivated by the astonishing potential of large language models (LLMs) for generating high-quality content in response to human language instructions, we embark on this work to examine the possibility of harnessing LLMs to automate AI accelerator design. Through this endeavor, we develop GPT4AIGChip, a framework intended to democratize AI accelerator design by leveraging human natural languages instead of domain-specific languages. Specifically, we first perform an in-depth investigation into LLMs' limitations and capabilities for AI accelerator design, thus aiding our understanding of our current position and garnering insights into LLM-powered automated AI accelerator design. Furthermore, drawing inspiration from the above insights, we develop a framework called GPT4AIGChip, which features an automated demo-augmented prompt-generation pipeline utilizing in-context learning to guide LLMs towards creating high-quality AI accelerator design. To our knowledge, this work is the first to demonstrate an effective pipeline for LLM-powered automated AI accelerator generation. Accordingly, we anticipate that our insights and framework can serve as a catalyst for innovations in next-generation LLM-powered design automation tools.