Web applications are on the rise and rapidly evolve into more and more mature replacements for their native counterparts. This disruptive trend is mainly driven by the attainment of platform-independence and instant deployability. On top of this, web browsers offer the opportunity for seamless browser-to-browser communication for distributed interaction. In this paper, we present Hector, a novel web application framework that transforms web browsers into a distributed application-centric computing platform. Hector enables offloading application logic to users, thereby improving user experience with lower latencies while generating less costs for service providers. Following the programming paradigm of Function-as-a-Service, applications are decomposed into functions so they can be managed efficiently and deployed in a responsive, scalable and lightweight fashion. In case of client-side resource shortage or unresponsive clients, execution falls back to a traditional cloud-based infrastructure. Hector combines WebAssembly for multi-language computations at near-native speed, WebRTC for browser-to-browser communication and trusted execution as provided by the Intel Software Guard Extensions so browsers can trust each other's computations. We evaluate Hector by implementing a digital assistant as well as a recommendation system. Our evaluation shows that Hector achieves lower end-user latencies while generating less costs than traditional deployments. Additionally, we show that Hector scales linearly with increasing client numbers and can cope well with unresponsive clients.