From my research, the web widget is not very useful even if you do follow Kay's advice and suppress it for users who are not logged in. We've spent a few weeks trying to get it working and will likely give up, due to its inability to work seamlessly. It has these issues:

1) Any submission from the web widget will show up with a warning saying "...was not signed in when this ticket was submitted" when the ticket is viewed by an agent.

2) The web widget doesn't allow you to remove the 'Email' field, so a user can edit that value to whatever they want, so there is no way for you to know whether the person who submitted the ticket is who they say they are. You can disable the ability edit that field (but it still looks like a field you can edit, which gives a bad user experience), but you cannot hide it.

3) The chat component of the web widget doesn't let you remove either the 'Name' or the 'Email', so is even worse.

It seems to me that not much thought was put into the design of the web widget to allow it to properly function when placed in a customer's own App.