The agentic web marks a structural transition from a human-centered information network to a digital environment populated by artificial intelligence (AI) agents that perceive, decide, and act autonomously. As delegated action unfolds at machine speed, exceeds discrete moments of human judgment, and distributes decision-making across non-human actors, existing legal frameworks face growing strain, creating an urgent need for new mechanisms capable of sustaining legality in this emerging order. A trustworthy agentic web therefore depends on the infrastructuring of legality through interoperable protocols that organize identity, delegation, and accountability across systems, enabling coherent governance beyond isolated platforms. Towards this end, this article advances a distributed legal infrastructure (DLI), a governance paradigm composed of five interlocking layers: (1) self-sovereign, soulbound agent identities; (2) cognitive AI logic and constraint systems; (3) decentralized adjudication mechanisms for dispute resolution; (4) bottom-up agentic market regulation to mitigate information asymmetries and network effects, including insurance-based models; and (5) portable institutional frameworks that enable legal interoperability while preserving plural sources of authority. This reference framework contributes to emerging research on embedding legality within agentic web infrastructure, aligning distributed technical systems with accountability, contestability, and rule-of-law principles.