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.