eyeores mom, the lots at the theme parks won't be closed if you go during mid-August. Attendance actually begins to drop off a bit at that time of year, since most of the southeast go back to school then (here in NC, August 9th). TL will be crowded at that time of year due to the heat, but it's open late too. Arrive after 3PM and you may be able to take someone's parking space who's left for the day. Or arrive right after the midday thundershowers. I'd hate to wait around for the TL bus back to our resort and we've ALWAYS driven to the water parks for that reason.
The one time that I've been at WDW without access to a rental car, I only went to the theme parks and back to my hotel. For that, Disney transport was fine.
With children, we have always used our car for anything else and sometimes for the parks (besides MK). Besides groceries and replacing broken sandals, ruinned swimsuit, etc., we've also had a couple clinic visits over the years (ear infections after all that swimming, strep throat, etc.). For us, bringing kids on a long vacation means having a car.
Besides, we all hate standing around when we are tired after standing in so many other lines, to wait for the return bus, or the kids' sitting on the sidewalk in the middle of the queue. And we take a day off from the parks, here and there, to enjoy our resorts. Those days usually include a character meal or nice dinner at a resort restaurant other than ours, as well as boat rentals. Getting there by Disney transport is too drawn out. We drive.
If it would make the difference between being able to go or not, affording a much-anticipated special dinner or something like that, then skip the car. Otherwise, I'd get it.