How can I connect 6 microphones for an annoucer system?

You leave it slightly vague and unclear where the announcement is coming from and where the announcement is going to.

I imagine a situation with a central admin station needing to communicate to six different places but only on a one place at a time basis.

To achieve that six momentary push button switches might be used to make the connection - the operator holding down the switch when speaking. Just 6 cable runs.

To have it the other way with any one of six places communicating to a central station is much more complicated - again you could use a push switch but it would need to be configured to disable 5 other stations when in use - each room in mutual contact with every other room and each room in contact with the central place - 36 sets of wire run.
A more complex and more expensive switch.

Good Luck...
September 2015