It sounds like you need to:
1. Open Chrome's settings page: chrome://settings
2. Click "Show advanced settings..." at the bottom
3. Under Privacy, click "Content settings"
4. Under Cookies, ensure "Allow local data to be set" is selected, and that "Block third-party cookies and site data" is NOT selected
5. Click Done
If you refresh the tab with the Web Reader and try re-opening it, it should now work.