When playing with such many players, especially when some of them are playing for the first time, we have one person holding one part of the box and collecting all the voting cards - when someone is ready he put it here.
Box is usually hold above the heads of all other sitting people (when playing with so many players there always is someone not sitting by the table so it is not the problem), to noone see the the answers. Usually the person holding the box is the one checking all the answers then, and tracking the score - it is more easy to sort and check all the voting cards in the box. Many of the players usually don't want to know the exact scoring mechanisms, so it is fine for them.
One disadvantage of this solution is when people stars to throw their voting cards to the box, inside of putting them gently. If they are doing so, sometimes, the pegs are moving out - anyway you just need to tell people to be more gentle.