Thank you everyone for your replies. So I can see that the short answer is: "Shadowing is necessary and volunteering or other clinical experience can't replace it."
The big question now is: How can I find shadowing opportunities? I have classmates who are pre-dental and they have been able to find shadowing opportunities during their freshman year by just calling dentists. Is that the same for physicians? However, I think there is a big difference here because most dentists have their own practice so they have a lot of freedom on deciding whether undergrads can shadow them or not, while most physicians work in a hospital setting with not much of freedom and strict restrictions. Who should I contact for shadowing? The physicians themselves or the hospital?
Contact:
1) individual physicians, aka "cold calling." I have never done this, so I have no idea about its efficacy. I don't think you'll end up needing to do this, though.
2) a medical school in your area, if you have one, which may help arrange shadowing for pre-med students, or have resources for you to utilize [my state school is UW, and they have a good list of resources here:
http://www.uwmedicine.org/education/md-program/admissions/applicants/pages/shadow.aspx#premed]
3) your state's Area Health Education Center, which will have resources to connect students from/interested in rural areas with physician mentors; Montana and Washington have specific contact people who help you arrange shadowing with rural doctors. Other states likely have good resources for this as well.
4) UW has an Alternative Spring Break program that arranges shadowing and week-long homestays with doctors. Plumb the depths of your school's pre-health advising center to find opportunities that you might not be aware of.
5) the Washington Academy of Family Physicians arranges mentoring for pre-meds, which can sometimes include shadowing your mentor at their practice. Dunno if other states have this, but it's worth looking in to, or getting in touch with your state's Academy of Family Physicians to see if they know about arranging opportunities.
That said, your best chance is to
use your network. The vast majority of my shadowing was arranged with the help of the physician who supervised my undergraduate research, who knew colleagues and former classmates in the area. I have shadowed at six major hospitals and several rural clinics, and none of them vetoed my participation. They were all quite helpful.
I also met a few doctors while I was in the process of shadowing their colleagues. Just be outgoing and show sincere interest in learning from your mentors. Ask your current preceptor if they know of any colleagues who might be willing to host a student.
None of this was nepotism. None of my connections were made from mommy and daddy calling their doctor friends. It's not impossible, so don't be discouraged. You just have to step outside your comfort zone and ask for help; be humble, outgoing, and show excitement.