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Some non-profit organizations would have to protect themselves from potential lawsuits by prohibiting minors from volunteering.
Non-profit agencies that are likely to have this policy include charities that cater to causes that involve potentially dangerous activities.
These agencies include (but are not limited to) animal welfare organizations, child abuse hotlines, battered women's shelters, legal rights advocates, drug/alcohol rehabilitation centers and suicide hotlines.
Many places would be all to glad to have you volunteer for them. My sons worked in thrift stores, food banks and at the Veteran's hospital. Possibly you might check the local humane societies or "no kill" shelters. The work may not be glamorous, but you will definitely take the experience with you. If you want to work at a local hospital next year (this is August), you should make the calls for an application now. Typically, those fill up quickly.