Black-box web scanners have been a prevalent means of performing penetration testing to find reflected cross-site scripting (XSS) vulnerabilities. Unfortunately, off-the-shelf black-box web scanners suffer from unscalable testing as well as false negatives that stem from a testing strategy that employs fixed attack payloads, thus disregarding the exploitation of contexts to trigger vulnerabilities. To this end, we propose a novel method of adapting attack payloads to a target reflected XSS vulnerability using reinforcement learning (RL). We present Link, a general RL framework whose states, actions, and a reward function are designed to find reflected XSS vulnerabilities in a black-box and fully automatic manner. Link finds 45, 213, and 60 vulnerabilities with no false positives in Firing-Range, OWASP, and WAVSEP benchmarks, respectively, outperforming state-of-the-art web scanners in terms of finding vulnerabilities and ending testing campaigns earlier. Link also finds 43 vulnerabilities in 12 real-world applications, demonstrating the promising efficacy of using RL in finding reflected XSS vulnerabilities.