Step-by-step Solution:

1. Identify the given equations. The equation of the circle is $S \equiv {x^2} + {y^2} - 9 = 0$. Its center is at the origin $(0, 0)$ and its radius is $r = 3$. The equation of the straight line is $4x - 5y = 20$.

2. Set up the coordinates. Let $P(x_1, y_1)$ be any point on the given straight line from which tangents are drawn to the circle. Since $P(x_1, y_1)$ lies on the line, it must satisfy the line's equation: $4x_1 - 5y_1 = 20 \quad ... (1)$ Let $M(h, k)$ be the midpoint of the chord of contact of the tangents drawn from $P(x_1, y_1)$. We need to find the locus of $M(h, k)$.

3. Find the equation of the chord of contact. The equation of the chord of contact of tangents drawn from an external point $P(x_1, y_1)$ to the circle $x^2 + y^2 = 9$ is given by the formula $T = 0$. $xx_1 + yy_1 = 9 \quad ... (2)$

4. Find the equation of the chord with a given midpoint. The equation of a chord of the circle $x^2 + y^2 = 9$ whose midpoint is $M(h, k)$ is given by the formula $T = S_1$. Here, $T = hx + ky - 9$ and $S_1 = h^2 + k^2 - 9$. So, the equation is: $hx + ky - 9 = h^2 + k^2 - 9$ $hx + ky = h^2 + k^2 \quad ... (3)$

5. Equate the two representations of the chord. Equations (2) and (3) both represent the same line (the chord of contact). Therefore, the coefficients of $x$, $y$, and the constant terms must be proportional. Comparing equation (2), $x x_1 + y y_1 = 9$, and equation (3), $x h + y k = h^2 + k^2$, we get: $\frac{x_1}{h} = \frac{y_1}{k} = \frac{9}{h^2 + k^2}$

6. Express $x_1$ and $y_1$ in terms of $h$ and $k$. From the proportionality, we can express $x_1$ and $y_1$ as: $x_1 = \frac{9h}{h^2 + k^2}$ $y_1 = \frac{9k}{h^2 + k^2}$

7. Use the condition that $P(x_1, y_1)$ lies on the line. Substitute the expressions for $x_1$ and $y_1$ into the equation of the line (1): $4\left(\frac{9h}{h^2 + k^2}\right) - 5\left(\frac{9k}{h^2 + k^2}\right) = 20$

8. Simplify to find the locus. $\frac{36h}{h^2 + k^2} - \frac{45k}{h^2 + k^2} = 20$ $\frac{36h - 45k}{h^2 + k^2} = 20$ $36h - 45k = 20(h^2 + k^2)$ Rearranging the terms, we get: $20(h^2 + k^2) - 36h + 45k = 0$

9. Generalize the locus. To find the locus of the midpoint $M(h, k)$, we replace $(h, k)$ with $(x, y)$: $20(x^2 + y^2) - 36x + 45y = 0$

10. Compare with the given options. The derived locus is $20(x^2 + y^2) - 36x + 45y = 0$, which matches option A.

Conclusion

The correct option is A.