Can someone be added to to be a authorized user on the AMEX card If he or she has no Social Security card number ?
Yes, but the card will be cancelled if the authorized user's SSN is not provided to Amex within 60 days.
Here's the screenshot of the form used to add an authorized user; it requires date of birth and SSN within 60 days.
I believe American Express will accept ITIN in lieu of SSN, but certainly one or other will be required within the 60 days.
Edit: And upon further review, MrCreditInternational is correct that passport numbers may be used as well to verify identity for AUs. That does require a call rather than inputting the information online, and they may ask for a photocopy of the passport.