To integrate with Google Docs API, please go to this documentation.
You can give manage permission of documents to non-admin team members from role settings (Settings > Access Permission > Roles). Admin users will always get access.
Then the permitted users will get a new menu item in the left menu. Labeled as Google Docs.
Note: If the user's left menu or default left menu is already edited, then the Google Docs won't be shown by default. Then you can add the menu manually from the left menu settings.