This sounds to me like they have not installed the sign in tool. The sign in tool identifies a user to SharePoint Online automatically; when you first log in and each time you access an Office document.
If you don't have the sign in tool, then your web browser and Office are behaving as expected: Office runs separately to the browser session so when you go to access a document, Office has to authenticate you.
You can mitigate this to some extent by checking the box that asks to save your username and password.
Ultimately, the process is easier if the sign in tool is installed.
Hope this helps.
Myles
http://www.thinkscape.com