just a note about the RAVS trips that ckgilabert mentioned - they are currently all filled up for the summer, so you won't be able to apply for any until they post the next round of trips, probably at the end of the summer.
if you have no tech experience, you cannot volunteer as a tech, but you can volunteer as an "other volunteer/support", which is what i'm doing this summer. from the descriptions in the training material, it seems like they often stick "support" personnel with the task of helping clients fill out forms, although i'm hoping i'll get to do more.
for the trip that i'm going to be doing at the end of may, there are 33 vet students, 4 DVMs, 2 vet assistants, 2 tech students, and i'm the only "support" person. so i'd say it seems more likely that you'd get selected to do these trips when you're actually a vet student.
in terms of volunteering/shadowing a vet, the way i finally started shadowing was by going to career services at my college, and getting the name of an alumni who is a local veterinarian.