Im 10 years old and my mom has been buying me sports bras. No I'm no flat I am developing some breasts but anyway so I'm basically that" girl with the breasts in my class." All the other girls are flat and my mom realized I was developing breasts when I was in 3rd grade so I started wearing sports bras but now they are popping out. My mom doesn't realize and I need teen/ padded bras... How do I ask?

You don't want a padded bra. If you ask for a padded bra she will say no because as a mother of a 10 year old she would not want your chest to look bigger than it is.
You probably should ask her to buy you a training bra. If you are bigger then you should ask for a teen bra. Tell her the sports bras just aren't working. I have seen some sports bras recently with a little more shape to them. Maybe look for one a little bigger and more of a bra support looking style.

Im going to make a leap here and guess that she has been buying your sports bras for you. So when the opportunity comes your way to go shopping with mom then pick out what you might like and just ask if you can get one of "these" ?

There is no magic to it. Just be sure you get one that properly fits. The first step in asking is to speak up.

Also if a sales clerk is working that department you could sort of get her attention and she might help make suggestions. Sort of side with you and get your mom to see.

I'm going to guess here that what is "popping out" are your nipples. When you started to develop your breasts became noticeable, but not your nipples, so sports bras were sufficient. And you want a padded bra so they won't stick out.

One answer is to wear looser clothing so you shape won't be as noticeable. But there are training bras meant to deal with this situation. There are also inserts that can help.

Just go to mom and show her how noticeable your buds are and tell her you now need something more than sports bra and ask when you can go shopping,