Vector search has emerged as the foundation for large-scale information retrieval and machine learning systems, with search engines like Google and Bing processing tens of thousands of queries per second on petabyte-scale document datasets by evaluating vector similarities between encoded query texts and web documents. As performance demands for vector search systems surge, accelerated hardware offers a promising solution in the post-Moore's Law era. We introduce \textit{FANNS}, an end-to-end and scalable vector search framework on FPGAs. Given a user-provided recall requirement on a dataset and a hardware resource budget, \textit{FANNS} automatically co-designs hardware and algorithm, subsequently generating the corresponding accelerator. The framework also supports scale-out by incorporating a hardware TCP/IP stack in the accelerator. \textit{FANNS} attains up to 23.0$\times$ and 37.2$\times$ speedup compared to FPGA and CPU baselines, respectively, and demonstrates superior scalability to GPUs, achieving 5.5$\times$ and 7.6$\times$ speedup in median and 95\textsuperscript{th} percentile (P95) latency within an eight-accelerator configuration. The remarkable performance of \textit{FANNS} lays a robust groundwork for future FPGA integration in data centers and AI supercomputers.