The single most important thing to do is to talk to faculty or doctoral students at your academic institution. They can help you find local opportunities, which are often not well documented or publicized. Other campus-level resources may give clear ways to engage. But most often, such resources are informational like this site, and document past research activities rather than future opportunities. Faculty (and doctoral students, if your institution has them) are the ground truth on what is possible at your institution.
There are many opportunities to engage in research at other universities. In particular, the National Science Foundation (NSF) maintains a list of Research Experience for Undergraduates (REU) sites, where faculty in a particular field have organized programs. CRA also organizes a Distributed Research Experiences for Undergraduates (DREU) program, connecting undergraduates with universities around the United States. Note that international students are eligible for the CRA DREU program, while the NSF REU program is open only to U.S. citizens.
Some universities have other programs that aren’t listed on the NSF site because they have a different source of funding. For example, human-computer interaction researchers at the University of Washington organize an annual summer research experience, but this is not an REU site. Once again, faculty are likely to be your best resource for discovering opportunities.
National laboratories are an often-overlooked as an opportunity for research-focused internships. With locations across the United States, national laboratories conduct basic and applied research sponsored by the U.S. Department of Energy. Other government agencies such as the National Security Administration (NSA) also sponsor internships with a research focus. The CRA provides a list of government and industry research labs.
There are many resources that provide an overview of research experiences, such as this CRA town hall webinar on getting involved.