See this thread regarding the difficulty with booking an appointment.
ireland/difficulty-booking-an-appointme ... 95033.html
Btw you should start looking for appointments 10 weeks in advance, not 4 weeks, although it is still very hard to find one by yourself.
Just walk in to the Burgh Quay office and ask to be seen on the same day or next day. Just say that her employer insists that she has a valid permit at all times. You can first try some quieter afternoons and if that doesn't work, you may need to go to the office in the early morning.
Legally she is not allowed to work if she does not have a valid permit even for 1 day. My fiancee had the same experience of having to be put on unpaid leave for 1 day because we couldn't find an appointment. She walked into the office that afternoon and was seen on the same day. Having said that, if her employer doesn't know when her permit expires, it is unlikely to be an issue for INIS or DBEI.
And yes it will impact her citizenship application. She must have valid permit for the last 365 days before she applies for naturalisation.