In the school context this comes down to the teacher librarian selecting the titles that are most suitable in content and interest for a range of readers of varying ages at that school. A good rule of thumb is the age of the central characters...isn't Bella 16 at the start of the series? That places these books well out of the context of the average 10-12 year old's experience. Also reflected in the movie adaptation's rating, which was M and not a PG.
MD, as a Teacher Librarian (TL for short) I pretty much stick to that guideline age of characters as to what age is likely to be okay to stock in the library. I have worked in a R-12 school setting and in 8-12 school settings. I actually ummed and ahhed over actually getting the series into my high school where I currently work (okay on mat leave from). I wasn't so keen on getting the whole series (I haven't read them - no time so far) but had heard/read reviews that meant I wasn't keen on getting book 4 in but the bookseller I got the first 3 from accidentally gave me book 4 instead of book 1 and I only realised after it was processed so ended up with the whole series. I was lucky in the end that it was just Yr 10-12s that actually wanted to read the series and had asked for it to gotten in. I wouldn't have allowed access to it by students under Yr 9 in the R-12 school that I worked at. They have separate collections for R-3, 3-9, 10-12. Students can read down but not up (the Yr 8/9s can if they have parental permission). I would have been putting the twilight series in the oldest collection.
I don't think it's appropriate for a primary school library.