I am using Mailbox myself for one of my sites, so I can tell you exactly what to do.
First of all, make sure that you are NOT using the Light plan. You need to be using the Standard or Premium plan.
If you have enabled Two-factor Authentication on the account of the Mailbox user who owns the Drive you MUST create an Application Password instead of using your regular password. To do that go to the hamburger menu next to the Mailbox logo, Settings, Security, Application Passwords. In the Choose An Application drop-down select WebDAV. Give the app password a name, e.g. "Akeeba Backup". Then click on the Add New Password button.
Now, here's the insane part they have not documented. When you connect to the root of your WebDAV storage you will see one or more directories whose name is the name of the user account who owns the drive. You need to know that and use it! If you don't, the upload will fail.
For example, when I connect to my Mailbox.org WebDAV storage with KDE Plasma's Dolphin file manager using the URI webdavs://dav.mailbox.org/servlet/webdav.infostore/Userstore I see a folder named something like "Nicholas K. Dionysopoulos". This means that in order to upload my backup archives to the backups folder of my Mailbox drive, the Directory I need to set up in Akeeba Backup is /Nicholas K. Dionysopoulos/backups. You can determine what that magic undocumented directory name is by connecting to your Mailbox.org drive using a desktop WebDAV client (on Windows and macOS I strongly recommend CyberDuck). You will see the name of this directory and note it down. From now on I will call it Your Full Name.
With that in mind, the configuration for your Mailbox.org drive using WebDAV in Akeeba Backup is as follows:
This is 100% fully and manually tested by yours truly. I am actually using it for a production site of mine. The two things you won't read in the documentation is the app password (which you figured out), and the magic directory name which you wouldn't have even thought about unless you had thought to connect to WebDAV from a desktop client. Now you know what I know :)
Nicholas K. Dionysopoulos
Lead Developer and Director
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