Choice D is correct. A point $\left(x,y\right)$ is a solution to a system of inequalities in the xy-plane if substituting the x-coordinate and the y-coordinate of the point for $x$ and $y$, respectively, in each inequality makes both of the inequalities true. Substituting the x-coordinate and the y-coordinate of choice D, $14$ and $0$, for $x$ and $y$, respectively, in the first inequality in the given system, $y\le x+7$, yields $0\le 14+7$, or $0\le 21$, which is true. Substituting $14$ for $x$ and $0$ for $y$ in the second inequality in the given system, $y\ge -2x-1$, yields $0\ge -2\left(14\right)-1$, or $0\ge -29$, which is true. Therefore, the point $\left(14,0\right)$ is a solution to the given system of inequalities in the xy-plane.

Choice A is incorrect. Substituting $-14$ for $x$ and $0$ for $y$ in the inequality $y\le x+7$ yields $0\le -14+7$, or $0\le -7$, which is not true.

Choice B is incorrect. Substituting $0$ for $x$ and $-14$ for $y$ in the inequality $y\ge -2x-1$ yields $-14\ge -2\left(0\right)-1$, or $-14\ge -1$, which is not true.

Choice C is incorrect. Substituting $0$ for $x$ and $14$ for $y$ in the inequality $y\le x+7$ yields $14\le 0+7$, or $14\le 7$, which is not true.