Soon to be Dr E is right. Academic centers are the best bets to get clinical research because the hospitals get a lot of research grants and the physicians there are usually heavily involved in research. Clinical research can be very variable depending on the studies being done. Most prospective randomized trials that involve trying new drugs or surgical procedures or imaging are going to be hard to do as a pre-med student because they usually will need someone with clinical experience. These types of studies which are the "latest and greatest" usually go to residents and sometimes med students. But there are many retrospective studies, case-controls, cross-sectional studies, and meta-analyses that are done on pre-existing data like chart reviews or medical databases that require extensive digging through previous records and charting and recording values or outcomes. These studies are often time consuming and few physicians have the time available to do these themselves, residents included. These are the studies that med students most often get involved in and what a pre-med could likely get involved in as well if he/she showed enough interest in it.
There are usually tons of research projects that are on hold just due to lack of manpower or people with available time. If you email or visit someone in the department and let them know that you are genuinely interested in that field and wanted to help out, I think it could be very realistic to get involved. Try emailing either one of the residents (often time they themselves are the PI in a study), an attending, or one of the department secretaries or coordinators that could connect you with one of the attendings to discuss your potential roles in a study
Many times, if you show enough dedication and put in a lot of hours, if you are working with a good resident or attending, you can be listed as second or third author on the paper if it is published and even accompany the resident to a conference if the paper makes it there. This is only if you are really lucky though. In either case, having your name on a clinical research paper even before med school is huge because these things are even important when applying to residency.
Good luck to all.