Integer grid points

Classes. Each lattice point has coordinate parities in $\{(\text{even},\text{even}),(\text{even},\text{odd}),(\text{odd},\text{even}),(\text{odd},\text{odd})\}$ — $4$ classes.

Pigeonhole. $5$ points in $4$ classes: two share the same class.

Midpoint. If two points have the same parity in both coordinates, their sum has even coordinates, so their midpoint is a lattice point.

Conclusion. Two points have a lattice midpoint.