It does vary quite a bit by state, but I think a lot of what causes the hinderance has to do with insurance/liability for the vet clinic. As I understand it, generally, volunteers are not covered under the clinic insurance.
Also, labor laws technically dictate that it's illegal for for-profit businesses to have "volunteers" do work that otherwise paid employees would do. This one isn't taken as seriously by most people, but it's out there.
non-profits is a completely different story though. so, spay/neuter clinics and shelters with a good vet services dept would be your best bet if you want to get hands on experience with technical skills as a volunteer. for example, as a volunteer at a shelter with a spay/neuter clinic, I was allowed to medicate, assist in emergencies, perform health checks/intake, draw blood, microchip, assist with xrays, intubate, prep for surgery, induce anesthesia, etc... otherwise, the best thing to do would be to get hired as a vet tech/assistant at a private clinic.
but honestly though, technical skills aren't required for vet school. what's most important is that you get quality time with practicing veterinarians, so don't despair if you can only find shadowing experience!