The epidermis is the outer layer of human skin. The layer underneath it is called the dermis. The epidermis functions as a protective layer over most of the human body, keeping out pathogens and unwanted chemicals. It also helps control moisture levels in the body by retaining moisture and releasing any excess via pores. It averages about 0.1 mm in thickness over most of the body, but it can be considerable thicker in more calloused areas like the heels and hands.
In the thickest parts, the epidermis contains five distinct layers: basal, spinous, granular, translucent, and cornified layers. It is classified as epithelial tissue that is composed primarily of a type of cell called keratinocytes.
The epidermis is void of blood vessels, including capillaries. Any coloration in the skin is due to pigment molecules like melanin. As a result, the epidermis cells receive nutrients via diffusion from the dermis (which does contain capillaries). So the answer to your question is blood vessels.