Okay, so I've been searching far and wide to figure out how to do the following: I need to edit my boss' iCloud contacts. Usually I log into her iCloud but she does not want me to do that anymore because I can see her e-mail, etc. Is there a way that I can access her iCloud contacts using my own e-mail address? I am open to suggestions about third party apps as well.
Note be careful here, while this answer works it can add a level of confusion and might have side effects that I didn't consider. I did try it tho, and it works.
Create a new Apple ID. Most easily done by going to http://appleid.apple.com from a computer browser.
On your bosses phone go to Settings > Mail, Contacts, Calendars and tap Add Account. Add the new Apple ID you just created. Once authenticated, turn off the switches for Mail, Calendars, Reminders and Notes-- leaving only Contacts turned on.
Now the phone has two sets of contacts, her old set and a new empty empty set. Only the original set is displayed tho. To change that open the Contacts app and click "groups" in the top left corner. Note that on this screen you can check one or both sets of contacts at the same time.
Now go to the computer and export all her contacts from the original Apple ID and import them to the new.
Once the new set is ready for her to use, make sure to turn off the old set using Contacts > Groups on her phone. Also go to Settings > Mail, Contacts, Calendars and in the Contacts section change the "default account" to the new group, so that any new contacts she creates on the fly will go to the new group.
To save confusion you could also remove all the contacts from the old Apple ID. Also open Settings > Mail, Contacts, Calendars and edit the original Apple ID turning the contacts switch off.
Now the contacts are stored under a separate Apple ID and you can log into that Apple ID on iCloud.com to edit them.
iCloud does not support delegated management.
You could potentially add the iCloud account to your computer and only enable functionality for Contacts, but you could go in and enable email at any time so it really isn't any privacy advantage for your boss.
This would be done in System Preferences, then under "Internet Accounts".