Assisting Seekers in Pursuit of Integration and Rapid Employment Act or the ASPIRE Act This bill addresses issues related to asylum applicants and migrants, including by directing the Department of Homeland Security (DHS) to provide employment authorization for eligible asylum applicants. Currently, asylum applicants are not entitled to employment authorization but may be granted such by the Department of Justice. Such employment authorization is valid until the applicant receives a final denial of the application, including any administrative and judicial review. The bill also shortens the waiting period for receiving employment authorization to 30 days from the filing of the asylum application. In addition, the bill authorizes U.S. Citizenship and Immigration Services to take actions to address the asylum backlog, including by hiring additional asylum officers. The bill also authorizes DHS to provide funding to state and local governments and nonprofit organizations for support services for migrants (e.g., food and shelter).