I have dark underarms..is there a mild over-the-counter deodorant i can buy to lighten them?

No: Deodorants only cover up or change the aroma. The color of the underarms area gets darker because of genetics (it's just the way the person is made), dirt on the skin (only in people who don't shower much), or an imbalance in the hormones. So, one should see a primary care doctor or endocrinologist to get an opinion on the color of the underarms areas.

See a doctor: You may have post-inflammatory hyperpigmentation resulting from irritation or shaving or friction from the armpits rubbing against each other. If you identify the underlying source you may be able to improve the problem. Using a topical bleaching cream may help. Laser hair removal helps the problem if it's caused by shaving or waxing. See a dermatologist for proper diagnosis and treatment.