Given the problem, let's assume the standard equation of the ellipse is $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 (a > b > 0)$;
By combining the equations, we get
$\begin{cases} \frac{y^2}{a^2}+ \frac{x^2}{b^2}=1 \\ y=3x-2\end{cases}$,
Eliminating $y$ and simplifying, we obtain
$(9b^2+a^2)x^2-12b^2x+4b^2-a^2b^2=0$,
Since the endpoints of the chord are $(x_1,y_1)$, $(x_2,y_2)$;
Thus $x_1+x_2= \frac{12b^2}{9b^2+a^2}$,
And since the x-coordinate of the midpoint of the chord is $\frac{1}{2}$,
Thus $x_1+x_2= \frac{12b^2}{9b^2+a^2}= \frac{1}{2}×2$,
That is $12b^2=9b^2+a^2$,
Which means $12(a^2-c^2)=9(a^2-c^2)+a^2$,
Therefore $2a^2=3c^2$,
Therefore $e= \frac{c}{a}= \sqrt{\frac{2}{3}}= \frac{\sqrt{6}}{3}$;
Hence, the correct choice is $\boxed{\text{B}}$.
Given the problem, the standard equation of the ellipse is assumed as $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 (a > b > 0)$; thus, by combining and simplifying the equations, we get $(9b^2+a^2)x^2-12b^2x+4b^2-a^2b^2=0$, and by using Vieta's formulas and the midpoint coordinate formula, we obtain $\frac{12b^2}{9b^2+a^2}= \frac{1}{2}×2$, thereby solving it.
This problem examines the application of the relationship between conic sections and lines, the method of undetermined coefficients, and the application of holistic thinking.