Q: Set up child account without credit card?

This should help: Create an iTunes Store, App Store, or iBooks Store account without a credit card or other payment method - Apple Suppor…

From : Family Sharing and Apple IDs for kids :

Before you begin, make sure that you're using a credit card as your iTunes Store and App Store payment method. To comply with child online privacy protection laws, you will use the CVV or security code from a valid credit card as part of providing your parental consent. If the card on file is a debit card or another payment method, you’ll be asked to provide a credit card before you can continue. After you create the child's Apple ID, you can change your payment method back to a debit card.

So no, it doesn't look like you will be able to create a child account without one.