That drove my father nuts when we were kids. Partly because his parents didn't allow it and so he and his sister and brother didn't fight much.
So growing up my father didn't allow it. I mean it was against the rules to argue or fight. If we did, we all got punished. If he heard us, we were told immediately to close our mouths and he would always say, "if you don't have anything nice to say then don't say anything at all." If we continued even a little, we all got sent to our rooms. He never tolerated it.
He would often sit us down and explain that in life we would have friends that would come and go, but our family would always be there as long as we treated eachother right. He told us we were blood and that we should love eachother. It sunk in because as middle school children and highschool kids we never fought. Maybe tease eachother once in awhile, but rarely ever argue. WE were like the three musketeers. To this day we get along great. The occasional disagreement, but we always let it go.
I have dear friends that were never taught that and were always allowed to fight and try to work things out themselves. Unfortunately they never did and to this day as adults they can't stand eachother. That's two different families. They can't spend five minutes in the same room.
If you don't talk to your children and teach them how to work things out, then they won't learn. And they will steadily grow up disliking eachother. They won't respect eachother and they will always fight. Sibling rivalry is serious and you have to teach them how to overcome it.
Explain that it is not allowed period. If they do it they will be sent to their rooms with no tv, no games, no books, no nothing until they can talk to eachother nicely.
If you know the older one likes to atagonize his sister, then make him stop. Tell him he is not allowed to even make faces at his sister no matter how much fun he thinks it is....because it will cease to be fun when he's punished everytime.
Tell you daughter that she has to learn to control her anger. That it is unacceptable to blowup over things. WE don't yell and scream and throw a tantrum because someone is annoying us. That is she can't just walk away, look away, or tell you that he's bothering her....then she will have to go to her room until she learns to control her emotions better.
Put your foot down now before it's too late.