How do I order a second titanium card?

Re: "I have a family member that’s going to be a participant on the account." Regarding the "going to be" -- you would first need to set her up as a participant. See: Set up Apple Card Family and add a co-owner and participants - Apple Support

Then, if they are over 18 they can order their own card. See the following from Set up Apple Card Family and add a co-owner and participants - Apple Support

What participants can do on a shared Apple Card

• They're not responsible for payments.
• They can view their own transactions and information.
• They can spend up to the credit limit on the account but might have an optional transaction limit set by an account owner or co-owner.
• They can immediately use their shared Apple Card and get unlimited Daily Cash on all of their own transactions.
• If they're 18 years or older, they can order their own titanium Apple Card.
• If they're 18 years or older, they can opt in to build their credit and be reported as an authorized user on the account.3

If they are under 18, you should be able to order a card for them by using the following instructions from Request and activate a titanium Apple Card - Apple Support

Order a titanium Apple Card for a participant under 18

If you're the account owner or co-owner of an Apple Card Family account, you can order a titanium Apple Card for any participants on your account who are under 18.

1. On your iPhone, open the Wallet app and tap Apple Card.
2. Tap the More button , then tap Card Details .
3. Tap the participant's name.
4. Scroll down to Titanium Card and tap Request.
5. Follow the instructions on your screen.