[FEATURE] Add example web application using Agent SDK with browser-based tool approval

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

The Agent SDK documentation and examples focus on CLI/terminal usage. All canUseTool examples use terminal input() prompts. All streaming examples print to stdout. There is no guidance or example for the common use case of building a web-based interface for Claude Code agents.

Third-party projects like The Vibe Companion demonstrate significant demand for browser-based Claude Code interaction. However, without official guidance, these projects resort to reverse-engineering internal protocols rather than building on the Agent SDK.

Proposed Solution

Add an example web application to the claude-agent-sdk-demos repository that demonstrates:

1. Backend: A Node.js/Python server using the Agent SDK with WebSocket connections to browser clients
2. Streaming to browser: Forwarding StreamEvent messages to the browser via WebSocket or SSE
3. Tool approval via browser: Implementing canUseTool that sends approval requests to the browser and waits for user response
4. Session management: Creating, resuming, and listing agent sessions via HTTP API
5. Basic frontend: A minimal React/vanilla JS chat interface showing streaming responses and tool approval UI

The example should be minimal (not a full product) but demonstrate the pattern clearly enough for developers to adapt.

Alternative Solutions

• Document the --input-format stream-json stdin protocol: This would let developers build web UIs by wrapping the CLI with bidirectional stdin/stdout. Less elegant than the SDK but works today.
• Document the internal WebSocket protocol: Not recommended -- this would commit to an internal API. The SDK is the right abstraction layer.
• Link to community examples: Point to well-built third-party implementations. Risk: they may use unsupported internals or become unmaintained.

Priority

• [ ] Critical - Blocking my work
• [ ] High - Significant impact on productivity
• [x] Medium - Would be very helpful
• [ ] Low - Nice to have

Feature Category

• [ ] CLI commands and flags
• [ ] Interactive mode (TUI)
• [ ] File operations
• [ ] API and model interactions
• [ ] MCP server integration
• [ ] Performance and speed
• [ ] Configuration and settings
• [x] Developer tools/SDK
• [x] Documentation
• [ ] Other

Use Case Example

A developer wants to build a team dashboard where multiple developers can launch Claude Code agents from their browsers, see streaming responses, approve tool calls, and manage sessions. Currently they would need to:

1. Read the Agent SDK docs (terminal-focused examples)
2. Figure out how to bridge canUseTool callbacks to WebSocket connections
3. Figure out how to stream StreamEvent messages to the browser
4. Handle session lifecycle via HTTP API
5. Build a tool approval UI without any reference implementation

An official example would reduce this from days of experimentation to hours of adaptation.

Additional Context

The Agent SDK streaming-output.md page already has a "Build a streaming UI" section, but it's terminal-only (uses process.stdout.write). The user-input.md page has thorough canUseTool documentation, but all examples use readline for terminal input. The pieces are there; they just need a web-oriented example tying them together.

The existing claude-agent-sdk-demos repo would be the natural home for this.