What would a system of coordinates be like if we did not have negative numbers? Would this system allow us to locate all the points on a 2-D plane?

Without negative numbers, we can only measure distances to the right of the origin (along the positive $x$-axis) and upward (along the positive $y$-axis). The coordinate axes would consist only of rays starting at 0, not complete lines.

This restricts us to the first quadrant alone, points with $x \geq 0$ and $y \geq 0$. Points to the left of the $y$-axis (second and third quadrants) and points below the $x$-axis (third and fourth quadrants) could not be described.

So, no; without negative numbers we cannot locate all points on the plane; only one-fourth of it (Quadrant I, together with the positive halves of the axes and the origin).