Exposing WebMCP Tools to External AI Agents

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I have a very general and superficial knowledge of WebMCP and how it works (mostly from working with the imperative API), so sorry if this is already covered by the docs or seems obvious 😅
From what I understand, WebMCP is a great way for AI agents within the browser to interact with websites. Websites expose specialized MCP tools on a per-website basis rather than a general "read_html" tool (for example) for all websites. So this works great for a built-in browser assistant (e.g. Gemini) but how are these tools supposed to be exposed to external agents, such as Claude Code or goose? Will there be some sort of protocol for this, will users have to install a Chrome extension or something..?

Thanks in advance!