Let points are $A=(a_1,a_2)$ and $B=(b_1,b_2)$ If $d(A,B)=1$ then such points don't exist, because circle with points $A,B$ has radius $r \leq \frac{1}{2}$ and there are not other integer points on it. If $a_1 \neq b_1$ and $a_2 \neq b_2$ then take $X=(b_1,a_2)$ and $Y=(a_1,b_2)$ then $AXBY$ is rectangle, so is convex and inscribed. If $a_2=b_2$ and $a_1<b_1+1$ then take $X=(a_1+1,a_2+1)$ and $Y=(a_1+1,a_2+1-b_1+a_1)$ Then $BX=AY$ and $AB=XY$ so points $A,B,X,Y$ lies on same circle And so for all points $(A,B)$ except points with $d(A,B)=1$ such $X,Y$ exists.