Emailing large files and folders using the links from the MyCloud Web is one of the significant features of My Cloud. However, when that site is not accessible there is no way to do this. It happens quite often.
Are you emailing from a remote location or from the same local network as the My Cloud?
For remote access, one option is to setup a VPN server on the local network and bypass using the MyCloud.com web portal by connecting to the My Cloud’s local network using VPN.
Generally one can attach files from the My Cloud using their email client’s attach file option. One may run into file size limits within the email application however. Helps to map Shares if you need quicker access to a local network My Cloud Share.
How to map a My Cloud or NAS device on Windows
How to map a WD network drive on a Mac
The unfortunate reality with having to use a manufacturer web portal to access a network attached storage device is one is at the mercy of that manufacturer and the portal up or down time. The single bay/single drive My Cloud have limited options for remote access. One could use the insecure option of FTP to let others access their My Cloud or to access their My Cloud from a remote location.