No Internment Camps Act This bill prohibits the use of funds, including previously authorized funds beginning one year after enactment of this bill, to operate or construct a family residential center, whether operated by U.S. Immigration and Customs Enforcement or by another governmental or nongovernmental contractor. The bill transfers funds obligated for family detention centers to the Alternatives to Detention Account (established by this bill) to implement the Family Case Management Program and additional community-based non-detention programs for alien families. The bill establishes, and funds through FY2024, the Emergency Fund for Asylum Seekers. The Department of Homeland Security may determine the number of alien families who have entered the United States seeking asylum in a fiscal year and based on specified percentage increases in such numbers receive amounts from the fund for the processing of such asylum seekers.