Preventing Online Sales of E-Cigarettes to Children Act This bill revises requirements regarding the sale and delivery of electronic nicotine delivery systems (e.g., e-cigarettes, e-hookah, e-cigars, and vape pens). Specifically, the bill subjects the sale and delivery of these systems to certain requirements, including tax, licensing, and labeling requirements; prohibits the sale and delivery of these systems to anyone under the legal age to purchase tobacco products; requires the U.S. Postal Service to implement regulations to prohibit the mailing of these systems; and requires the National Institutes of Health to conduct a study on the health impacts of e-cigarette use by youths and young adults. The bill does not apply to products approved by the Food and Drug Administration for tobacco cessation or any other therapeutic purposes.