Do US REU programs sponsor the student visas (J1 or F1) for international candidates?
All official participants in NSF REU programs must be US citizens or permanent residents per the NSF REU homepage:


  Undergraduate students supported with NSF funds must be citizens or permanent residents of the United States or its possessions.


A self-funded student or non-NSF REU funded student would simply be an intern or student at the host institution.