Why do I get itchy around the chest area when I haven't exercised in a long time? Is it because of my tight sports bra or something else?

Exercise itch. My best guess is that it is suggest the sweat, which may or may not be obvious. It doesn't have to drip off you to cause an itch. If your in good shape (exercising regularly) you produce less sweat than when you are out-of-shape or deconditioned; in this case your body is more stressed by lower amounts of exercise. I assume the sweat is after you stop. If so, then just shower immediately.
Urticaria maybe. There is a entity called exercise induced urticaria. It is itching and hives induced by strenuous exercise. The bra rubbing may also be a factor. Try a Claritin (loratadine) over the counter about 1 hour before you exercise and see if it still occurs. Of course if there is any breathing difficulty, or wheezing associated with it, you should be checked out before further strenuous exercise.