In Team Fortress 2, is it possible to move more than one item at a time while sorting my in game inventory? I want to sort all of my items, but doing it one at a time feels very inefficient.
With the game, no. Every item must be moved individually to another space. You can batch-select items but only for deletion.
There is some external software provided at http://www.jengerer.com/item_manager/ which connects to your steam backpack and gives you more options, including one which allows you to batch-move. Maybe this is what you want.