With SharePoint Online 365, the first issue you might get with InfoPath forms is this error message below:
The feature requires SharePoint Server 2010 or greater with InfoPath Forms Services enabled:
To resolve this issue, you must have access to SharePoint admin center. Here are the steps involved:
1. In the Admin center, go to SharePoint admin center.
2. Select infopath menu.
3. Check “Allow users to browser-enable form templates“, and “Render form templates that are browser-enabled by users“, and then click OK.
4. You will get prompted with the message below, but just ignore the message and click OK:
Office 365 will support InfoPath Forms Services through 2015, but Microsoft is in the process of retiring InfoPath-based technologies. To learn more, please refer to http://aka.ms/ipfs Do you still want to enable InfoPath Form Services
This message only means that Microsoft is eventually depreciating and will no longer support InfoPath Form Services in the future.
5. Go back to InfoPath Forms, and you should now be able to connect to the SharePoint Online 365 list (also known as apps), or create a new SharePoint list.