We investigate the use of Web agents that use planning to achieve goals involving preconditions on the dynamic Web. We implement our approach, AEPIS, based on Semantic Web technologies and rely on environmental information evaluated exclusively at runtime to take dynamic changes into account. We evaluate AEPIS in static and dynamic environments, based on the Web of Things, and compare its performance to a rule-based and an agent using large language models. We find that AEPIS combines adaptivity with rapid planning at scale, even in a dynamic environment.