If a person were born without eccrine glands, what skin function would he or she have a hard time completing?

The answer is thermoregulation

Human is a warm blooded animal so they have to control their internal temperature. Eccrine glands is the synonym of sweat glands. Sweat has an important role to release heat as they will wet the skin, allow it to evaporate in huge area. If the sweat glands doesn't work, the person will have difficulty to dissipate heat when the body temperature increases.