It's very possible to get more clinical (non-clerical) volunteer opportunities, but you may have to go beyond the posted volunteer information. Here's a couple of strategies:
1. Get to know a doctor by shadowing them (to do this, you call or e-mail a bunch of doctors at your university or community hospital) a few times, then ask if you could volunteer on their service (for instance, if you're shadowing a pediatric neurologist, you might be able to spend a few hours up on the neurology unit a couple of times a week) or go on ward rounds. Usually, these doctors are on service only a fraction of the time, so you may have to wait for their turn to come, or alternatively they can introduce you to another physician who is on service. Also, if you shadow for long enough, you'll often wind up in a 'volunteer' like position (running for drug samples, helping patients get to their MRI, getting the results of a test, etc.).
2. Start out working in a clerical job. Odd as it sounds, clerks really are needed. Once you know people, you can talk to them directly about what you'd like to see/do. Most residents are happy to have you around -- it gives them a chance to talk through stuff, and in the process of teaching it to you they both feel good about themselves and it helps cement things in their brain.
3. Get an EMT license. I've never done it, but I know they're eligable for more hands on work.
4. Get a job at the local hospital in patient transport. Again, something I've never done, but some of my friends who did it for extended periods have talked a lot about it. Basically, it's a lot of scut, but you get to meet patients and see stuff while you're waiting for the patients.
5. Contact the volunteer office directly. Tell them what you want to do; express an understanding that what they really need help with is clerical work, but you'd also like some time to spend reading to the kids, etc.
6. Find out if there's a "Child Life" office or something similar. These offices deal with improving quality of life of kids at the hospital. Usually, they administer programs like reading to kids, playing with them, etc.
7. Seek out alternative (i.e., outside the hospital) volunteer opportunities like the one Hamface 16 was talking about. Nursing homes, rehabilitation centers, camps are all good ones, but the latter two might be more promising given your pediatric interest. Aside from onc camps, there are epilepsy camps, diabetes camps, etc. Also, check to see if there are any mentoring opportunities in your area for kids with chronic diseases -- they tend to have very hiigh incidence of self destructive behavior, teenage pregnancy, etc. so there are often support groups to help them deal with the chronic illness.
8. Get in touch with local non-profits that deal with disabled children. Small non-profits often have volunteers take more active roles (for instance, after demonstrating commitment, you might be put in charge of a project; or you might be able to suggest a project that they then let you run with).
9. There are programs in most cities for bringing food to the homebound, many of which specialize in a particular population (elderly, AIDS, etc). If you get involved in one of those, you might be able to speak eloquently of the social cost of disease. I remember one young man I met in clinic who said his only regular contact with other human beings was through the local food delivery non-profit. Keep in mind that these are jobs that require sustained commitment (not necessary a lot of time, but it needs to be very regular).
10. Call a home visit nursing service and see if you can tag along.
Hope that helps.
Best of luck,
Anka
