Food Stamp Integrity Act of 2017 This bill amends the Food and Nutrition Act of 2008 to revise the rules for disqualifying individuals from the Supplemental Nutrition Assistance Program (SNAP, formerly known as the food stamp program). The bill permanently disqualifies individuals who have been found guilty of offenses or misconduct, including: fraud or misrepresentation with respect to SNAP; the trading of a controlled substance, firearms, ammunition, or explosives for benefits; terrorism; defrauding the federal government, a state, or a unit of local government; obtaining cash by destroying food purchased with benefits and collecting deposits for containers; or selling food purchased with benefits. An individual who is delinquent in paying child support may not receive benefits unless a court allowed a payment delay or the individual is complying with a payment plan. If a household requests a third replacement Electronic Benefit Transfer (EBT) card in a one-year period, a state may: (1) require the head of the household to appear in person for an interview before issuing an additional card, and (2) terminate participation of the household for one year if a fifth card is requested in a one-year period. The bill modifies the work requirements and eliminates certain exemptions and waivers from the requirements. If an able-bodied adult without dependents fails to meet the work requirements for 3 months in a 48-month period (36 months under current law), they are ineligible for SNAP for the remainder of the 48-month period, or until they comply.