Appalachian Communities Health Equity Act or the ACHE Act This bill prohibits the authorization or continuation of certain mountaintop removal coal mining pending the results of a study on the health effects of such mining. Specifically, the National Institute of Environmental Health Sciences must study and publish the health effects on surrounding communities of surface coal mining that involves explosives and that is conducted in the steep slopes of Kentucky, Tennessee, West Virginia, and Virginia. Federal and state agencies may not authorize or renew any such mining until and unless the study concludes that there are no health risks; entities that currently conduct such mining must monitor and report pollution levels associated with such mining until the study's publication. The Office of Surface Mining Reclamation and Enforcement must collect a one-time fee from each entity that currently or previously conducted such mining to cover the federal cost of implementing the bill's requirements.