Do cosmologists see length contraction in the stars? Wouldn't we be able to see length contraction in objects, like stars, that we can see and are moving at relativistic speeds?
 A: In principle yes. But there are at least two reason that we don't observe this:
Stars are slow
First of all, stars don't move at relativistic velocities. Typical velocities of stars in a galaxy is of the order of $100$s of $\,\mathrm{km}\,\mathrm{s}^{-1}$. In rare cases, such as if one star of a binary goes supernova and gives the other star a kick, you can have "hypervelocity stars" that reach $\sim\!\!1000\,\mathrm{km}\,\mathrm{s}^{-1}$.
Galaxies themselves move with similar velocities. In the case of two galaxies merging, you can also have substantially larger velocities (of $1000$s of $\,\mathrm{km}\,\mathrm{s}^{-1}$), but this is still nowhere near relativistic velocities.
Stars are small
Second, stars are very far away compared to their sizes, so only the nearest ones ($\lesssim1000$ lightyears) are actually resolved, and since their angular sizes are measured in milliarcseconds, measuring a difference in the diameter of the same star in two directions would be extremely difficult. Moreover, since velocities of nearby stars tend to move in more or less the direction as we do, their velocities relative to are more like 10s of $\,\mathrm{km}\,\mathrm{s}^{-1}$.
Observed length contraction
So, how do we know length contraction is a real thing? Measuring it directly is not really possible, but several experiments show the effect anyway. For instance, muons created high above Earth's surface decay so fast that they wouldn't be able to reach the surface. Yet we detect them, and the reason is that in their reference frame, the trip to the surface is much shorter than in our reference frame, so they have the time to get down after all. Another example is protons in accelerators which are flattened due to length contraction. This cannot be seen directly, but can be inferred from the way they scatter.
Length contraction in an expanding Universe
EDIT: I understand now that you were thinking of length contraction of galaxies carried away from us because of the expansion of the Universe. This is actually a great question, which at the moment doesn't seem to have a good answer, but according to astrophysicist Tamara Davis, the simple answer is that the galaxies' recession velocities are not "peculiar" velocities (i.e. motion through space), and therefore we don’t see length contraction. I may be able to update this answer in some months.
