You are going to need to work with her, because you are dependent upon what is in the house, and what she buys.
I scarcely think all she buys is meat. I bet you've got ice cream, the makings for salad and potato chips in your home. Those are all vegetarian, but only one of them is actually good for you.
The point I am making is, you need to educate yourself about nutrition before you attempt to make the case to your mother that you want to be a vegetarian. She needs to know that your growth won't be stunted, etc. Might I suggest SparkTeens
(yes, it's a diet site) because it's free and comprehensive, and it will teach you about nutrition. I have little doubt that there are teens on there who are vegetarians or are considering it.
Also ask to see your pediatrician, as s/he will have suggestions, and it will show your responsibility in this area. I became a vegetarian when I was 13, and was so for a good 15 years (I still don't eat beef, but I did go back to eating chicken, and I was always eating fish, eggs and dairy products back then), and my doctor was more concerned about iron for me, than protein intake, and I used to take a supplement. So do this, and demonstrate that this is a responsible decision you're making, and not just an "I hate meat" decision.