4. The major axis of the ellipse is 6, the left vertex is on the circle $(x-3)^{2}+(y-2)^{2}$ $=4$, and the left directrix is the $y$-axis. Then the range of the eccentricity $e$ of the ellipse is ( ).
(A) $\frac{3}{8} \leqslant e \leqslant \frac{3}{4}$
(B) $\frac{1}{4} \leqslant e \leqslant \frac{3}{8}$
(C) $\frac{1}{4} \leqslant e \leqslant \frac{1}{2}$
(D) $\frac{1}{2} \leqslant e \leqslant \frac{3}{4}$
4. A.

Since $a=3$, let the left vertex be $(m, n)$, the center of the ellipse is $(m+3, n)$, and the $y$-axis is the directrix, $\frac{a^{2}}{c}=m+3$. Therefore, $c=\frac{9}{m+3}$, and $e=\frac{c}{a}=\frac{3}{m+3}$.

Since the vertex lies on the circle $(x-3)^{2}+(y-2)^{2}=4$, we have $1 \leqslant m \leqslant 5$, so $\frac{3}{5+3} \leqslant e \leqslant \frac{3}{1+3}$, which means $\frac{3}{8} \leqslant e \leqslant \frac{3}{4}$.