I am an owner of a small hotel, I have been informed today that the housekeeping staff are planning to 'make' the new front of house person leave because they don't like her pointing out things that they are not doing, which they are supposed to be doing. How do I deal with it? If this does happen can I take disciplinary action and under what category?

You have a staff meeting. Without revealing what you heard tell the staff that times are tough and everyone has to step up and do their jobs. Anyone that is not pulling their weight will be replaced.