Hi Erin, thanks so much for your question! I have to say bus routes are not my strong-point since being a resident an hour away, we always have our car to drive. However, I put my research Mouse Ears on (aka I asked my fabulous fellow Moms & Dads for help!) and looked into an answer for you and your family.
You actually have a few options. The Disney buses start service an hour prior to the opening time of the park they are headed to. You could take a bus to Disney's Hollywood Studios or Epcot and either take a bus or boat from DHS, or walk or take the boat from the International Gateway in World Showcase in Epcot. However, the entire trip could take around 45 minutes to an hour with wait time for the buses and boats. If Epcot or DHS has Extra Magic Hour mornings on the day of your ADR, you'll be fine, but otherwise, you might be re-enacting scenes from the Amazing Race as you rush to breakfast!
Another option, though not free, is a taxi service. It would only be about a 10 minute trip that way, and probably cost about $10 according to the
Mears taxi estimator site. With an ADR that early, it might be the best way to ensure you make it there on time and not worry about missing your dining reservation.
I hope this helps with your planning, please let us know if you have any other questions and we'll be happy to share our thoughts! ~Suzannah