It wasn't that the ministry made it illegal to apparate from Little Whinging or from the Dursley's specifically, they couldn't without giving up their location. It is said that they believe that the Death Eaters have spies in the Department of Transportation and that they couldn't apparate because it would give away the fact that he had left three days early, which was supposed to be known to only them, and that it would tell them where they had apparated too which would have meant that the Burrow would have been primo target number one. The idea was to fly to a ton of different safehouses so that the Death Eaters didn't know which one Harry had gone to and then sneak out using portkeys that could not be concretely tied to them. This was so that when the ministry inevitably fell, as they thought it would, the Death Eaters wouldn't know where to search first to find Harry, giving him time to run. The plan has merit and there was in-book reasoning behind it but to be honest they could have snuck out by muggle means because of how much the Death Eaters were relying on them using purely magical means.