A study comparing a group of chronically depressed individuals with an

Premises:
People who are depressed are found to have more immune system disorders.

Conclusion:
Mental states influence the body’s vulnerability to infection. (In other words, depression causes immune system disorders)

Doesn't this make you think that it could be the other way around too (in which case the conclusion would be wrong)? Just because depression and immune system disorders appear together, it doesn't necessarily mean that depression causes immune system disorders. What if actually immune system disorders often cause depression? You will see similar results if you study depressed people vs others in that case too.

Anyway, let's look at the options now to find the one that weakens.

Option (C) is right on target -
(C) Disorders of the immune system cause many of those individuals who have them to become chronically depressed.

(E) A high frequency of infections can stem from an unusually high level of exposure rather than from any disorder of the immune system.
On the other hand, (E) gives you an alternative cause of immune system disorders. Note that this does not weaken our conclusion. Just because there are additional ways that lead to immune disorders, it does not mean that depression may not be responsible for immune disorders.

Taking a simple example:
If we see that A and B appear together, can we say that A causes B? Well, that's not necessary. B could cause A instead.
What if I tell you that C causes B? Does it mean that A cannot cause B? Absolutely not! Perhaps both A and C can lead to B. So knowing that C causes B, does not weaken A causes B.
Hence (E) is not correct.