I think it is wonderful that are so concerned for your sister’s well being but you also have to understand that you are not the parent and can’t control the situation. You do have to let go of what you don’t have power over. It will only continue to upset/stress you and everyone around you. YOu can try to persuade her, but not force
You can help educate them. You can suggest her mom read some books like In Defense of Food or see Food Inc. or Jamie Oliver’s Food Revoloution. You can suggest your sister eat vegetables, but cannot force her to. Try making games or challenging her— “I bet you can’t eat x many pieces of broccoli!”. Make it fun, not tense. You can also make some meals yourself— especially some that contain the vegetables in them (stews, curries, soups) that can’t be separated. Make some smoothies that have lots of fruits and vegetables in them.
Stay a good example to your sister. Make sure there are vegetables at each meal. Discuss health choices. And for now, just don’t push, let it happen on its own. If this were your own child I would advise differently. Hopefully if and when you have your own, there won’t be these issues at 10— you’ll have started them off the right way.
Best of luck.