The US records fingerprints of foreign visitors using a system formerly called US-VISIT, now known as the Office of Biometric Identity Management. One might think that this is done under the authority of 8 USC, Chapter 12, Subchapter II, Part VII, Registration of Aliens.
However, a closer look shows that Title 8 and associated regulations do not describe the operation of OBIM. For example, at 8 CFR 264.1(e)(2), fingerprinting is waived for essentially every nonimmigrant who remains in the US for less than a year:
(e) Fingerprinting waiver.
(1) Fingerprinting is waived for nonimmigrant aliens admitted as foreign government officials and employees; international organization representatives, officers and employees; NATO representatives, officers and employees, and holders of diplomatic visas while they maintain such nonimmigrant status. Fingerprinting is also waived for other nonimmigrant aliens, while they maintain nonimmigrant status, who are nationals of countries which do not require fingerprinting of United States citizens temporarily residing therein.
(2) Fingerprinting is waived for every nonimmigrant alien not included in paragraph (e)(1) of this section who departs from the United States within one year of his admission, provided he maintains his nonimmigrant status during that time; each such alien not previously fingerprinted shall apply therefor at once if he remains in the United States in excess of one year.
I can attest from personal experience that people falling under paragraph (1) are not fingerprinted at the border. However, despite paragraph (2), most other nonimmigrants are fingerprinted at the border.
Under what authority is that fingerprinting performed?