Sorry, only admins can create and edit families. Is is not possible to allow a user to create and edit families because any user could then add any other entry to their family.
Here’s a link to a FAQ which explains how to create a family. So you do not have to re-add addresses and phone or any common data, instead of Adding a New Entry, create the first. After you save that entry mouse hover over it and click the copy action. It’ll copy the entry info into a new entry.
Hope that helps!