[FEATURE] Support Claude Chrome extension integration in VSCode extension

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

I use the Claude Code VSCode extension, and installed the Claude Chrome extension, but am unable to use the two together, I have to choose between one or the other.

Proposed Solution

Support integrating with the Chrome extension from VSCode

Alternative Solutions

_No response_

Priority

High - Significant impact on productivity

Feature Category

Other

Use Case Example

_No response_

Additional Context

_No response_