There is probably a software of some sort online that is a random sorter. I know for a fact that a friend of mine has a software that you insert names into, and it randomly picks as many as you need(out of a hat, too. It's kind of cool. :'D) which might work. Like regarding who would be on each team, the cabin names could be picked out of a hat and the first however many are on team one and the second however many are on team two. Then every person who wants to participate signs up, and there names are put into the system. The first person whose name is drawn from the system gets to capture the flag for their team? It could be like a week or two long rp or something, but the only person who truly gets to capture the flag is the person who was chosen, so we know in advance.
Maybe a rule can be set up that anyone who was NOT the chosen person who writes that they captured the flag AND brought it across the line and won the entire game doesn't get to participate in the next one? Just to deter anyone from doing it and claiming to be the winner?
And of course put in the other obvious rules, no killing/maiming/injuring another character without the owner's consent, no god-moding or power-playing, etc.
And a thread could be made under announcements or something that is updated each time a game is played saying a small congrats to the winning team with a list of which cabins were on that team during that particular game?
Organizing a game of capture the flag would be difficult, but I definitely think it's possible with some thinking.