I'm sixteen as well (seventeen in three months! ) and I work at a vet office in the boarding kennels and dog bathing/grooming section of it. I got the job when I was 15, but it was because of connections in the right places. (My dad helped teach the boss' daughter in college for Pharmacy.) They haven't regretted it though, and I'm actually the only employee in the kennel/bathing section that is still around from when I originally started!

The most simple thing you can do is ask. If a big chain petstore (Petsmart, Petco, etc.) won't take you on, then go to a smaller local feed store. Go to a grooming parlor and ask if they have any job openings for a bather. Go to a veterinary clinic and ask if they would be able to start you off on some odd jobs, and then work your way up to working in the kennel portion (if they have boarding).