You can ask, but be prepared to be turned down. Some restaurants might do it. There are reports that the Wave will. As far as the parks, Le Cellier might on occasion allow guests to carry out soup, but not a full meal. Via Napoli allows no carryout orders at all.
You can definitely carry out sushi at any time from the Kona sushi bar at the Polynesian.
I do know that Olivia's at OKW will not allow any carryout dining at all. They used to, but they stopped it after the dining plan came into being. The story was, apparently, that guests expected to take out their food from Olivia's and pay with a counter service credit rather than a table service credit, so they just stopped allowing carryout.