Hi Julie,
I'm sure you are going to have a great vacation with your family! And good for you for looking for the most relaxing route from place to place. Vacation should be more of a stroll than a race.
From Epcot, there isn't a way to connect to the Downtown Disney
water taxis that I'm aware of. But that doesn't mean you should miss out on the scenic ride from the International Gateway area (between the United Kingdom and France). From this alternative exit to Epcot, you can take the boat to Disney's Hollywood Studios, passing the Boardwalk, Yacht and Beach Club resorts and the Swan and Dolphin.
Expect that ride to take about a half hour. At Hollywood Studios, you can catch a bus for a quick ride to Downtown Disney.
Or, if you find yourself closer to the main entrance of Epcot at departure time, you can simply catch a bus straight to Downtown Disney from the main Epcot bus station. Cast members can help point you to the right spot.
Have a relaxing visit with your family, and please let me know if I can answer any other questions.
Wilma