Aviation Industry Assistance for Cleaner and Quieter Skies Act This bill directs the Department of Transportation (DOT) to establish an Aviation Industry Assistance for Cleaner and Quieter Skies Voucher Program to provide electronic vouchers to air carriers to offset the purchase or cost of a lease of new fuel-efficient aircraft in exchange for commitments from such air carriers to decommission certain currently-used aircraft and sell the aircraft for recycling of parts or disposal. DOT must register aircraft recycling firms that are eligible to purchase currently-used aircraft and establish requirements and procedures for the recycling of parts or disposal of such aircraft to ensure that such aircraft are taken out of service and not used to develop other aircraft with higher greenhouse gas emissions.