You're describing behaviour in terms of a momentary pushbutton switch. This makes it harder to keep track of whether each person is available or not. I suggest you use either a toggle switch or a latching switch instead. Connect the switch between the digital input and ground, and enable the internal pull-up resistor. It would be nice (but not essential) to debounce the inputs. It would be a good idea to provide feedback for each input station so it's obvious what their status is being reported as. You can get latching switches with internal LEDs to show the status, or you could add an external LED to a latching switch or toggle switch.
The simplest way I can see to keep track of your list would be to number the inputs as you have done, and store the available numbers in an array. When a number becomes unavailable, loop through the array to locate the entry containing that number, and shift all the subsequent entries down one to remove that entry from the array - finally, zero the last entry (which is now unused). When a number becomes available, append it to the array.
It would be sensible to have a sanity check to make sure the number is not already in the array, for extra resilience
If you're using the serial monitor to display the status, then any time the status of any input changes you would print out the status of all entries.