[FEATURE] Add 'Claude in Chrome (beta)' support in 'Claude Code for VS Code'

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 VS Code extension does not currently support the /chrome command like in Claude Code CLI. So I am unable to control the browser with Claude Code when using the VS Code extension.

Proposed Solution

I'd like to be able to use the /chrome command when using the VS Code extension.

Alternative Solutions

_No response_

Priority

High - Significant impact on productivity

Feature Category

CLI commands and flags

Use Case Example

_No response_

Additional Context

<img width="320" height="144" alt="Image" src="https://github.com/user-attachments/assets/f9eeb04d-140f-4278-a7f3-8ccc6c81a4bc" />