Yes, Chase explicitly says that the AU cards will be reported: Chase provides account information to the credit reporting agencies for all account users. This information could impact an authorized user’s credit score. When you tell us to add a user to your account, you’re confirming that you have a relationship with the person or people whose name(s), address(es), and date(s) of birth you’ve told us, that all their information is correct, and that you have their consent to add them. If Chase determines you’ve given us fraudulent name, address, or date of birth information or did not have such consent, Chase can close this account.

https://www.chase.com/personal/credi...uthorized-user

Capital One also does not collect SSN for Authorized Users, and the credit bureaus match to the correct person.

In any event, if the cards are not reported correctly you can contact Chase later and ask to have the SSN added.