Chris,
This may constitute unwanted advice but I'd not deal with both these issues at once.
Your church can set whatever vacation time seems reasonable. But it seems to me that punishing the secretary for having taken vacation she was given may be a bad idea here. That is how it will look if you talk about reducing days off and the emergency call system at the same time. The fact that no system was set up for you to get a call when she is on vacation isn't her fault unless she was tasked with creating such a system. I would deal with both issues separately.
I would deal with the emergency contact issue now. Then down the road, when this cools off a little, I'd ask whoever constitutes your HR committee to deal with what is or isn't appropriate days off around Christmas. If the call had come on the 24th would you be saying that the secretary shouldn't get off Christmas Eve etc.? Because given your current setup the same thing would have happened on the 23rd, the 24th or even the 25th (5 days, 4 days or three days with no visit respectively.)
It is the system of emergency contact (or lack thereof) that is at fault and not a staff member getting a day off given her by the church. And the family's fault in part for waiting five days before trying something again.
(End potentially unwanted advice.
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