REUs are not institutionally limited.
There are two types of REU grants. One type is just this side of automatic for researchers with a current NSF grant. These are called REU Supplements, and typically will support up to two undergraduates in your lab.
The second type of REU grant is a site REU. This program is designed for a research group to develop a research experience for undergrads that will be mentored collectively by the research group. Typically the students take part in a summer experience and are recruited on a national basis.