Accountability Through Electronic Verification Act This bill expands the E-Verify program by requiring all employers to use it and permanently reauthorizes the program. Currently, E-Verify use is voluntary for most employers, although some states mandate its use. All employers must use E-Verify to confirm the identity and employment eligibility of all recruited, referred, or hired individuals, including current employees who were never verified under the program. Failure to use E-Verify shall create a rebuttable presumption that the employer is violating immigration law. U.S. Citizenship and Immigration Services must generate weekly reports about individuals who have received a final nonconfirmation of employment eligibility. The Department of Homeland Security (DHS) must use the report to enforce immigration laws. DHS must establish a program to help certain small businesses verify employee eligibility. DHS shall also update E-Verify's design to help prevent and detect fraud and identity theft. The bill increases civil and criminal penalties for hiring non-U.S. nationals ( aliens under federal law) who are not authorized to work. DHS must debar repeat offenders and those criminally convicted from holding federal contracts, grants, or cooperative agreements. The Social Security Administration, Internal Revenue Service, Department of the Treasury, and DHS must jointly establish a program to share information to help identify non-U.S. nationals who are not authorized to work. The bill establishes the Employer Compliance Inspection Center within Homeland Security Investigations of U.S. Immigration and Customs Enforcement. The center's duties include processing I-9 employment eligibility verification forms and ensuring compliance with employment eligibility laws. DHS must report to Congress on ways to simplify procedures relating to I-9 forms and on whether the I-9 process should be eliminated.