Q: How do I set up a family apple account without a credit card?

I don't think so. Both of these support documents mention the need for a credit card to finish the setup of child accounts so that there is a central billing. Family Sharing - Apple Support    and this one Family Sharing and Apple IDs for kids - Apple Support

Mommy2MJs wrote:

 

I don't have a credit card so it won't let me set up my kids an AppleID.  Is there any way to bypass this?

You need a credit card at initial setup.  This is a legal requirement that you, as the Family Organizer, are taking financial responsibility for the minors.  After it is set up, you can switch to a debit card.  But there have been no exceptions to having the credit card initially (and not for lack of people asking).