Creating a new family member's vault and password?

I signed into the family account on the family members computer. 
...
But when I log out and back in, 1P wants the family vaults password?

The process for inviting members to a Family account does not involve signing in to a Family Organizer's account on the member's computer.
You should clean up the stored data for 1password.com on (the profile of the web browser used by the account on) that computer, send the member an invite, then help them follow the instructions to sign in to their 1Password account using their computer (and any other devices).

Note that vaults do not have passwords.

I created a new vault in the family account and moved their passwords to the new vault

When signed in, the family member should have their own Private vault. If you set up the access to the "new" vault correctly, then they should be able to move the items from it to their Private vault. It would be wise to subsequently remove that "new" vault to avoid any confusion.

Hi @silsurf, thanks for writing in!

As AJCxZ0 shared, the first step to adding family members to 1Password Families is inviting them through a 1Password app, or on 1Password.com. Once the invitation is received, your family members will be prompted to set up their own account within 1Password Families. They will get their own Secret Key, and will use their own account password (which can be the same as they used for their individual account). By default each family member has a private vault, accessible only to them, and a shared vault which is accessible to everyone in the family. Everyone with an account can create more shared or private vaults to help manage and share credentials. 

Bellow are the steps your family member can follow to migrate from their individual account to the family account. I’d recommend following these steps on a computer where they’re using 1Password for Mac, Windows, or Linux already. If you or your family member are unsure about anything, or you’d like a hand diving in deeper to any specifics, please feel free to contact the support team at support@1password.com

1. Accept the invitation you sent, and create a new account within 1Password Families.
   * I would suggest having the Emergency Kit and account credentials for both accounts on hand.
2. Add their family account to a 1Password app where they're using the individual account.
3. Move or copy the existing items in the individual account to the family account.
4. Sign into their family account on 1Password.com and confirm the items are now all available there. If the items aren't available stop here!
5. If they have no intention of using the individual account, they can delete that account. Otherwise, they can keep the individual account and cancel the subscription.

Let us know if you have any questions, and thanks again!