Solution:

Suppose the vectors sum to $s$. Take the $x$-axis along $s$ (or in any direction if $\mathbf{s} = 0$). At each move the first player picks the vector with biggest $x$-coordinate. Each player makes $996$ moves and the $x$-coordinate the first player picks on any move is larger than the $x$-coordinate the second player picks on the following move. So the sum of the first player's choices has larger $x$-coordinate than the sum of the second player's. Since the sum of all the $x$-coordinates is non-negative, the sum of the first player's choices must also have larger absolute value. The sum of the $y$-coordinates of all the vectors is zero, so the sum must be the same for the first and second players. Hence the first player's sum has larger magnitude than the second player's.