Family Sharing Verification Code Problem

After recently changing the default payment method, family sharing members were prompted for a verification code. The code was sent to the main account holder, but the dialog box would disappear before the family member could enter any code.

SOLUTION:

Family member had to go into the payment settings in *their* own iCloud settings:

Settings > Apple ID > Payment & Shipping > verify

This ended the verification prompts and they could complete the purchase.