Currently, there is no native option within iOS Settings or iCloud to export your Hide My Email aliases, labels, and notes directly to a CSV or JSON file. The only official method to obtain a structured copy of this information is to visit https://www.google.com/search?q=privacy.apple.com, sign in with your Apple ID, and select "Request a copy of your data"—specifically choosing iCloud Mail—though be aware that this process can take up to seven days for Apple to prepare the download.
For a faster but more tedious solution, you can manually copy the details from the Hide My Email settings at iCloud.com/mail, but if you require frequent exports or advanced management, you may want to consider switching to a dedicated service like SimpleLogin or AnonAddy, which are designed with native export tools for power users.