Chemical Peel for Hyperpigmentation Around the Eyes?

Other less aggressive treatments should be considered first, including skin bleaches (hydroquinones, etc.), microdermabrasion, and Triluma. A chemical peel may work if the coloration is due to the melanin or dark pigmentation from the pigment forming tissues. If the color is due to blue colored veins beneath the thin skin, or deep set eyes that sit in a shadow from the orbital rims, none of this will be effective.
Get it evaluated carefully, and listen to all of the options before deciding which is best for you.

You may actually have melasma (hormone-related darkening of the skin). Before you consider any treatment options, I would suggest you visit a dermatologist to get an accurate diagnosis - there are many excellent ones in the city of San Francisco.
Generally speaking, for pigment problems of the face, light chemical peels can be beneficial, but treatment selection also depends on your skin tone, since the darker the skin tone, the greater the risk for darkening of the skin with peels. You do, however, have options including hydroquinone for skin lightening.