docs: add VS Code install badges (#532)

Closes #531

(Once https://github.com/mcp is the default GA registry in VS Code, the
link can be updated to show the registry page)

---------

Co-authored-by: Alex Rudenko <alexrudenko@chromium.org>

Author: digitarald

README.md

@@ -157,12 +157,22 @@ Configure the following fields and press `CTRL+S` to save the configuration:
 
 <details>
   <summary>Copilot / VS Code</summary>
-  Follow the MCP install <a href="https://code.visualstudio.com/docs/copilot/chat/mcp-servers#_add-an-mcp-server">guide</a>,
-  with the standard config from above. You can also install the Chrome DevTools MCP server using the VS Code CLI:
-  
-  ```bash
-  code --add-mcp '{"name":"chrome-devtools","command":"npx","args":["chrome-devtools-mcp@latest"]}'
-  ```
+
+**Click the button to install:**
+
+[<img src="https://mcpbadge.dev/badge-install-in-vs-code-stable-dark" alt="Install in VS Code">](https://vscode.dev/redirect/mcp/install?name=io.github.ChromeDevTools%2Fchrome-devtools-mcp&config=%7B%22command%22%3A%22npx%22%2C%22args%22%3A%5B%22-y%22%2C%22chrome-devtools-mcp%22%5D%2C%22env%22%3A%7B%7D%7D)
+
+[<img src="https://mcpbadge.dev/badge-install-in-vs-code-insiders-dark" alt="Install in VS Code Insiders">](https://insiders.vscode.dev/redirect?url=vscode-insiders%3Amcp%2Finstall%3F%257B%2522name%2522%253A%2522io.github.ChromeDevTools%252Fchrome-devtools-mcp%2522%252C%2522config%2522%253A%257B%2522command%2522%253A%2522npx%2522%252C%2522args%2522%253A%255B%2522-y%2522%252C%2522chrome-devtools-mcp%2522%255D%252C%2522env%2522%253A%257B%257D%257D%257D)
+
+**Or install manually:**
+
+Follow the MCP install <a href="https://code.visualstudio.com/docs/copilot/chat/mcp-servers#_add-an-mcp-server">guide</a>,
+with the standard config from above. You can also install the Chrome DevTools MCP server using the VS Code CLI:
+
+```bash
+code --add-mcp '{"name":"io.github.ChromeDevTools/chrome-devtools-mcp","command":"npx","args":["-y","chrome-devtools-mcp"],"env":{}}'
+```
+
 </details>
 
 <details>