Why doesn't the light from galaxies appear stretched?

Galaxy rotation happens at a very slow rate (compared to the speed of light). Let's suppose you are observing a galaxy edge-on that the delay from the farthest point is $\Delta t = d/c$, where $d$ is the galaxy diameter. If we take the lag from one extreme point to the other as D:

$D = vt = \frac{v}{c}t$

(where $v$ is the rotational speed).

You can see that, to have an appreciable effect, your velocity must be a fraction of the speed of light. The Milky Way rotates at a speed of $\sim10^2$ km/s, which is $\sim10^{3}$ times less than $c$. This means that your lag from one point to the opposite one of the galaxy is exceedingly small.

Of course, it is possible that for larger galaxies (see, e.g., Hercules A) you can have faster rotation, but you still need a coincidence of factors like distance, angle, and size of the galaxy, to appreciate a small difference in the lag. And this, as far as I know, has never been observed yet.

Galaxies would appear stretched along the line of sight, not jumbled. Let's say a galaxy (specifically, the closest side of it) is ten million light years away and, as you proposed, is 100,000 light years across and we see it nearly edge on. The front of the galaxy will appear to us as it did ten million years ago and the back of the galaxy as it did 10,100,000 years ago. Thus, if the galaxy is moving towards us it will appear bigger than it actually is due to the delay between light from the front of the galaxy and the rear of the galaxy reaching us. If the galaxy is moving away from us, it will appear smaller than it actually is (compressed along the line of sight).

As far as the effects of this time delay on viewing the rotation of the galaxy, the length of the cosmic year (the length of time it takes for the sun to travel around the center of the Milky Way) is 225 to 250 million years. So, as many other people have already pointed out, the rotation period of a galaxy is small compared to the time it takes for light to travel across the galaxy and there is no jumbling effect.