Take a look at Colgate University. If you look at Health Sciences and then Beyond the Classroom you will see opportunities for shadowing and internships. I have a family member who graduated and is now in medical school.
Many students also find shadowing opportunities at their pediatrician and local teaching hospitals (if you live near any).
My daughter attended a public university with a medical school and hospital on campus. They had a medical mentors program that assigned premeds with current medical students who they met with 1-2 times a month to help guide them through the process and answer questions. That might be something to look into.
My daughter volunteered at the hospital on campus. She had to go through a series of interviews and pass a test in order to be certified in the language she was going to speak at the hospital. I said this earlier, but I think it is important to ask the school about their policies regarding on campus hospital volunteering. It might not be an automatic green light for all premeds/prehealth students.
I have a family member who attended the University of South Carolina Honors College. Although she was not premed, I will tell you that the honors college opens up a whole bunch of opportunities within a smaller college setting.