Hi there, KAMatlock.
Are you restoring a backup to your new iPad from a previous device? If yes, is it possible that you may have downloads or purchases that were made with your children's Apple ID email addresses? We're asking this since you mentioned it keeps asking for the passwords to their Apple ID email addresses.
Can you also tell us what you see when you see these prompts? If there is an option to cancel or not enter that now, what happens when you choose that?
We want to make sure we fully understand what might be going on, which your answers may help with that.
In the meantime, we do want to let you know that only one person should be using an Apple ID. Family Sharing allows everyone to have their own Apple ID but still share applicable purchases or subscriptions. If you'd like to learn more, the links below go into further details:
Take care.