Each person, regardless of their role (family owner, manager, or member), can only access their own personal vault. This means that nobody has the ability to view or access another person's vault within the LastPass Families subscription, even if they are the owner or a manager.
While you cannot share your whole vault with another person, you can use the Sharing Center to share individual items (passwords, payment cards, secure notes, and so on), or create a shared folder and fill it with items to share with another member of your LastPass Families subscription.