You can generally be a vet assistant once you're 16 and legally able to work.
To become a licensed vet tech, it's usually a couple more years of schooling beyond high school to get your certification and pass the exam, so I would think you would most likely be 20+ before it would be possible. Any vet assistant could call themselves a 'vet tech' if they're not licensed, and some really do have the same skills that take a few years of good training to get, so again you're looking at 20+ minimum.
I have to add something to this, because it's a rather large peeve of mine, since I have completed my associate degree, taken my national exam, and am awaiting the chance to take my practical state exam. Bear with me.
Yes, any assistant could call themselves a tech, but it's actually ILLEGAL to do so. Furthermore, it's misrepresentation. Sure they may "have the same skills" but just because you can draw blood, put in a catheter or administer medication does not mean you have an understanding of anatomy, physiology, disease processes, and treatments. Just food for thought!
OP, I think it's wonderful that you want to be a tech! I did an accelerated program that only took 16 months. That's up to you, of course, but if you graduate highschool at 17, and went to school right after, you could feasibly be working as soon as age 19. Good luck, let us know how it goes!