This is using the Mercury browser on the iPad.
You need to use a computer or you could try an alternate web browser on the iPad, but you have to download one that will identify as a desktop browser. I have two browsers that identify as desktop browsers, iCab Mobile and Mercury. If you identify as a desktop browser in the settings, you can sign into iCloud.com on the iPad. However, in my experience I have found it to be a bit slow and not the greatest way to access iCloud.com. Some users swear by this though.
I do not use iCloud email so I don't know how well it will work for what you want to do, but you can try it and. These are paid versions of the browsers that I am talking about. Lite versions will not give you the full functionality of the paid versions.