While it's true that you can't use the keychain in Chrome, I can suggest a solution that I've made.
The explanation involves some web dev, so skip to the end if this is too boring.
The extension has two parts:
The extension generates a QR code and you scan it with your regular iPhone camera.
It opens the static website (hosted at GH pages) with the necessary parameters in the query string.
Because the website is static, it receives no data.
The QR code URL also contains a one-time pad encryption key. This key is passed through the URL hash.
You use the Keychain on your iPhone to enter the password, which is encrypted using the key and sent to the extension using WebSockets. The key is unusable after that. The built-in Safari functionality allows you to save the password for later reuse.
You can get it here: WinchPass - iCloud Keychain for Chrome