Can I use a glycolic chemical peel for hyperpigmentation on legs?

I have been using all kinds of alpha hydroxy peels since 1984, on all areas of the skin, including the legs, without any problems. During this time, I have found that concentrations below 50% do little for the skin. For this reason, for more than twenty-five years, I have used full-strengh (70% is the maximum concentration) glycolic acid peels. In properly selected patients, these peels, when performed in a series of six to eight at twice monthly or monthly intervals, work well for clearing folliculitis, fading postinflammatory hyperpigmentation, "sun" spots (solar lentigines), as well as for other indications. They can also contribute to an all around smoother and more lustrous appearance with essentially no downtime, when performed properly.Patient satisfaction tends to be extremely high, and superficial peels have stood the test of time. They have, in fact, recently making a strong comeback in spite of all the marketing hype and glitz for lasers and other energy-based devices. Daily, at-home use of topical glycolic acid products in lower concentrations may further enhance and complement the fading process and accelerate the pace of improvement.Consultation with a board certified aesthetic core physician is important for choosing the best approach for you.

You could, but you could also risk further injuring the skin with a chemical peel in areas that are already hyperpigmented. Injury to the skin, especially Asian, Brown, East Indian or Black, is prone to hyperpigmenttion after injury, bites, surgery, sclerotherapy ...
Sometimes, it takes up to 3 cycles of skin growth, 180 days each, for the hyperpigmentation to fade, if at all, without adjunctive therapy:

There are some topical creams that we use for patients with hyperpigmentation from a myriad of causes in the legs: from venous insufficiency, following wounding, following surgery, following sclerotherapy ...
We recommend Sclerovase and Scleroquin plus topical creams.
You can add retinoid RR Complex Retinoid Repair Cream as well. 
Scler-x post inflammatory hyperigmentation relief complex is a great supplement to take as well. See link below. 

Glycolic peels can be used on the legs but in terms of efficacy, I would not recommend them. In order to get significant results with post-inflammatory hyperpigmentation ("dark marks") on the thick skin of the legs, I would recommend deeper peels such as the Rejuvenize peel (a modified Jessner's peel) (safe of any skin type or color) or a TCA peel (best used on medium to lighter skin types). Several treatments are required (6-12 depending upon the severity of the hyperpigmentation) but a 50-80% improvement can be seen.