I've never done it at Magic Kingdom but we've done it a ton of times at Epcot. Son would get nachos at Mexico, husband would get a cheeseburger at America, and we'd all eat at China where I would have beef fried rice and egg rolls. Norway didn't understand us at first when we wanted our open faced sandwiches to go but they did get it after a while. The counter service at Japan does to go as well.
The all time "I can't believe they did that" was when a friend wanted us to bring her back a turkey leg from Disney World. When my husband told the kiosk person that, the turkey leg was triple wrapped in foil. They were going to put it in a bag but we already thought ahead and had a large Ziploc ready to go.
You shouldn't have any problem at all.