Any way to add grouping levels in Cloud Vault?

There aren't any such restrictions on M-Files Cloud in general. Is this a common view or a personal view that you are trying to modify? If it's a common view, does your user account have enough administrative rights to modify these views (either "Full control of vault" or "Manage common views and notification rules")?

Sorry, not sure I understand your response.  Is it possible to create grouping levels within common views in the Cloud version?  

If so, I will check with my IT department, as I thought I had full admin permissions for this vault, but don't see how to modify the common view - or even how to create a new common view.  Right-click menu is not the same as it is in the desktop version.  Should the right-click menus be the same? 

Sorry, not sure I understand your response.  Is it possible to create grouping levels within common views in the Cloud version?  

If so, I will check with my IT department, as I thought I had full admin permissions for this vault, but don't see how to modify the common view - or even how to create a new common view.  Right-click menu is not the same as it is in the desktop version.  Should the right-click menus be the same? 

Sounds like you are using the web interface. You need to use the Desktop interface, I believe.
Joonas' remarks are related to whether the server is in M-Files Cloud or on your own server.

Yes, this is correct. View creation and modification needs to be done with M-Files Desktop at the moment. M-Files Web will get this capability later this year.