Yes, K., talk to your doctor. Ask her to check you to make sure you are okay for tampons. Then ask her to show you how to use them and tell you what you have to be careful of and why. It will be a little embarrassing. I think that your grandmother and your mother call them "gross" because putting in a tampon entails touching yourself and knowing how your body is made down there. That's why they mention "married women".
Maybe you can't convince your mom. But you can use tampons. I would do that. That might not be the most popular answer here - teens should obey their parents in most things. But you are NOT living in yesteryear... sports and pads do NOT mix. I doubt that your mom and grandmother ever did sports when they were girls.
Your doctor should tell you not to leave in a tampon too long. They won't get lost - just make sure you bear down like you're having a bowel movement and pull on the string to get it out. Don't buy extra-absorbent ones because of the worry of toxic shock syndrome. Use a little vaseline on the end of the tampon to help it go in. You can find the place that it goes (the vagina) by peeing on a mirror and using the hole UNDERNEATH where you pee. (You have 3 holes - where you pee from, the vagina, and where you poop from.)
It might hurt to put in a tampon, but that's a bit normal the first time. You have to push it past your pubic bone, at an angle. The doctor can show you AND tell you which tampons to try. Ask the doctor to write it down for you. Don't try to do it before you go to the doctor.
After all this is done, THEN ask the doctor to tell your mom that your body is fine for using tampons. You might be surprising your mother, and she might not like it. But you have a right to talk to the doctor about ANYTHING, and you have the right to privacy about what you are going to say to the doctor. It would be nice if you can talk her into letting you use tampons. If she says no, go to your dad. Move heaven and earth to get your way on this. And if heaven and earth don't move, well, your call, K.. Just make sure you listen to everything your doctor says, read the insert in the box so that you know what you are supposed to do and not do, and make sure you comply.
Good luck.