Has Supergirl ever gotten sick before? According to this question, Superman has gotten sick before, but has this happened to Supergirl, and what are the effects?
In the TV show, yes.
Kara has longed to feel normal. Well, now she gets her chance. The fight with Red Tornado last week drained her powers, leaving her vulnerable to everything that plagues us mortals. For instance? Colds! Supergirl? More like Snifflegirl.
Cat is none too pleased that her assistant is the one sneezing in the office. “You never get sick, that’s the best part about you,” she says. She shames Kara into leaving.
Supergirl, being Kryptonian like her cousin, cannot get sick under normal earthly conditions, due to the Sun's yellow rays boosting her immune system with the rest of her Kryptonian body.
She can, however, be affected with Kryptonian viruses and get sick when she is weakened by kryptonite or something similar, or when she is outside of the Sun's rays (i.e. Phantom Zone).
Long answer: As a Kryptonian she is immune to all earth born diseases under a yellow sun in the comics. It would not make any sense just to make a comic with super girl having the cold/a runny nose ,who would want to read that? Maybe under a red sun radiation they could write a comic where she gets sick but the writers would not do that. In the TV show(which i find surprisingly accurate) she does get sick after doing her solar flare she then basically became human for a few days. Even then she should not have gotten sick due to her Krytonian DNA. If she were to get sick then I would imagine that the symptoms would be similar to those human deal with. The only weaknesses i known krytonians are vulnerable to are Kryptonite,Electricity(which they are pretty resistant to for example a bolt of lightning might feel like a taser to them.) and magic which they have a human level of resistance to.