I have received a request where I got the task that the people swap places so that they get to know each other better.

One possibility is that under a chair a major win is quoted in an envelope. All specators must stand up and search a new chair. One is the winner and has to open the envelope and on the paper is written that the price is that he has to assist on the stage for the next trick.

Do you have any idea how that "place changing of the spectators" could be integrated in a magic show?
There are about 70-80 spectators.

I can't answer that magically, but there is a game out there (you could probably find it on the internet somewhere) which is intended as a great mixer.

The thing is, you have to know information about each individual prior to the game. Odds are, they could send this information to someone by email prior to the actual event. Then, you list of all these little known facts about each individual on a piece of paper. One by one folks have to go around and ask each person a question to see if they can guess which questions (I played the violin in school, I met a famous movie star, etc.) goes with which person.

The first person who can get 5-10 right answers by associating the correct person to the right answer wins a big prize.

I've done this game numerous times and it's one of the best ice-breakers. Again, though, it has nothing to do with magic.

Gerry