First you need to find out what the legislation says in the country/county you live in... that will greatly depend on if you can work and/or volunteer, if so what hours/days and what you require to do so...eg in some places you may need your schools written permission, your parents written permission and/or an offical work/volunteer permit. once you know that then 'normally' small businesses are more likely to employ/allow volunteering than larger businesses as if you are employed or volunteering both need you to be covered by business insurance and larger organisations are more like to want school leavers as they can work longer hours than teenaagers still in school, unless your school/education authority supports work experience weeks, but 'normally' that is for those 1-2 yrs from finishing formal education, not your age group.
So small practice vets, or other animal practitioners like dog trainers, behaviourists , hydro, small independant pet shops, kennels/cattery/stables, however where ever is local to you go and ask , even write up basic CV to leave with them even if they say no, you are gaining experience of job interviewing which will help you the next time and leaving a CV with your details on and school grades, hobbies, likes and interests they may have a place later on and could contact you.
The employment law ( also covers volunteers/work experience) where I live a 13-14 yr old can work a maximum 12 hours a week, not before 7am or after 7pm, no longer than 2 hours on a sunday or 5 hrs on a saturday, also no job advert can state age offered to...so it is very difficult for a business to advertise for or discriminate against ( what they want/need).
I have one 14 yr old who came and spoke to me ask if she could come in volunteer and get experience, she came in first for a couple of hours on a saturday ( and only after I spoke to her parents and covered her with employment insurance) two weeks later I spoke to her parents again as I wanted to know how it was helping her( or not) from their perspective and as education is very important to me I would have also spoken to her school, but because of covid schools were closed, but I did phone the Head as soon as I knew they were getting ready to reopen and he was in school.... a month after she started I asked her if she wanted to do more hours and get paid for it as she is very good again I said 'depends on parent/school approval' .... I had a phone call from her Head about a week ago who tells me her grades have improved and she is a more confident and mature person, that really pleases me and her parents are thrilled with her...I love her she is a real credit to herself and a great colleague
I agree with everyone else. If you start volunteering at an animal shelter, wild life sanctuary or even zoo, then it will be much easier to get a job there after you learn the ropes of things and get to know the staff. If they really like you as a volunteer, then they will probably want to hire you.
It’a a similar thing with libraries. If you volunteer and donate your time to helping out because you genuinely love books, you’ll have a lot better chance of getting hired to be a librarian.