If you have some adults that can help you with the legal and financial aspects, then yes, you could definitely help start a volunteer organization as a teen.
You can apply for jobs or volunteer positions at veterinary clinics, ASPCA shelters, and boarding kennels. As a minor, you may not legally be able to work with or touch...
Many state agencies and some national parks have entry-level seasonal positions that do not require higher education; by starting at this level and working you are likely to gain experience...
Pick volunteer work that will help you spend more time with your peers. This might be through a club at school, a religious organization, or a community organization.