Hi, Richard! If you have more than one adult in your traveling party, I would consider bringing two strollers. Anyway you configure a double stroller -- side-by-side, front-to-back, sit-and-ride, -- they are cumbersome through the walkways, restaurants, shops, attractions and resorts of WDW. Besides, they're heavy when you have to fold and unfold them as much as you have to do to load transportation or to visit attractions. It will be much easier to maneuver if you put the 4-month-old in the stroller to which s/he is accustomed and put the 4-year-old in a smaller, lightweight umbrella-type stroller. Plus, the baby is likely to sleep much more in the stroller and for longer periods of time if the older, more excited "roommate" isn't bouncing around!
There are a few restaurants that may have the room in which to set a double stroller at your table, but there aren't many. However, a smaller single stroller would be welcomed at most restaurants. I hope this helps you plan! Have a wonderful day and a magical vacation!