Dining at the
Akershus Royal Banquet Hall inside Epcot's World Showcase does require a Magic Your Way ticket.
Any time a guest enters one of the major parks (Animal Kingdom, Epcot, Magic Kingdom and Hollywood Studios) whether it is to tour the attractions and/or visit a restaurant, admission is required.
If your Magic Your Way tickets include the park hopper option, your family would be able to hop to any of the other parks listed above.
To maximize your touring, I would plan on having this breakfast on a day that your family does plan to tour Epcot. In that case, you would not need the park hopper option unless your family wanted to go to another park in the evening.
Each time that you enter a turnstile, you'll need an admission ticket. You can then exit and enter that park as much as desired that day. With a park hopper ticket, you could visit more than one park in a single day.