As powellrj commented, a car is definitely not NEEDED, but...
Honestly, it kind of depends on where you are staying. The biggest hassle by far is getting between resorts. Since WDW buses only go from individual resorts to the parks, if you want to go from a value or moderate resort to a deluxe for a meal or character dining or event, it can take almost an hour from door to door between waiting for your bus at your resort, driving to one of the parks, transferring and waiting for the bus from the other resort, then traveling to the other resort...and then do it in reverse after your visit. The monorail resorts are far the easiest to get between, as are the clump of those next to EPCOT.
The other issue is if you want groceries, but then again, there's always wegoshop.com.
But whatever you decide, relax and enjoy your vacation!!
-R