Everything is open that week, except Christmas day, of course. Make sure to search for restaurants that might be open on Christmas day and reserve early. Definitely use up the entire week on Sao Miguel.
I don’t know of any houses that would fit that many people, but take a look at Quinta de Santana near Ribeira Grande-https://www.booking.com/Share-YBHahL. There are many 2 bedroom apartments on the grounds. I have stayed here and fully recommend it.
The location is good for exploring by car, but nothing is within walking distance, unfortunately. Perhaps they would prepare Christmas dinner for you, so ask.
There are lots of beautiful hotels that might interest you as well. Similar to the above link, search booking.com.
If I hear of any houses that could accompany your family, I will send you a private message.