I am usually late with my period, a week late at this point, but I am showing no signs of it coming. Could it be because I am sick or stressed?

Yes, but. Your cycle being late could be due to stress; it could also be due to pregnancy, so it's a smart idea to do a pregnancy test, just to be sure.
Yes. Periods can fluctuate with fatige, time travel, stress, pregnancy (check it a few times), age(menopausal?), medications, illness.
Late period. While, of course, you have to consider the possibility of pregnancy, as you guessed, both stress and significant illness can throw your hormones out of balance and result in delayed or missed periods. In these cases you often end up with irregular, unpredictable, intermittent bleeding, that can be treated with hormones.