To find the length of the side of the polygon between the two points \((7, -18)\) and \((7, 18)\), we can use the distance formula for points in a vertical line since both points have the same x-coordinate.
The length (distance) between the two points can be calculated as the difference between their y-coordinates:
\[ \text{Length} = |y_2 - y_1| = |18 - (-18)| = |18 + 18| = |36| = 36 \]
Thus, the length of this side of the polygon is
\[ \boxed{36} \text{ units}. \]