Disclaimer: this is just speculation from me, there are not (necessarily) explicit references in the books.
Probably every Noble House possessed some books used by their Maesters.
We can speculate that almost every House, surely all the Great ones but maybe also the most important of the Minors, kept at least a small collection of books, if not a proper library, in their home castles.
We know that every Major House employs the services of a Maester from the Citadel, whose main duties were of a teacher, a medic, a keeper of the ravens, an astronomer, and generally being an advisor about economics, warfare, and everything else that was related to knowledge.
Shortly, they were generally scholars and savants, not specialized on a single discipline but resembling more a polymath from our own Renaissance.
Every Maester very probably required books to do their work.
As we know from their collars, Maesters were required to master many different subjects, and when they left the Citadel to serve on the keep of their new lords, very probably they needed a (large or small) selection of books to use as an aid and reference to carry out their duties. According to what we know of their work, we can suppose that this set of books comprised text about:
* Astronomy
* Medicine
* Zoology and Biology (for the Ravens, but not only)
* Economy
* History
* Warcraft
* Genealogy and Heraldry
* Grammar and general literacy
Probably, great libraries were very rare.
We know that in Westeros books were not printed (movable type was not invented) but they had to be written and copied by hand.
This made books a relatively rare commodity, so if we can suppose that the most common books, those that we can compare to the "handbooks" and "manuals" that Maesters used for their work were fairly widespread (maybe an apprentice was required to copy some of these to carry with him, if he was sent to a House that did not possess that particular book?), other works (the most specialized ones, concerning more pure research and scholarly knowledge, and less about common "everyday" subjects for a Maester) were surely available in fewer copies and only in the greatest and most important libraries. When you have to copy by hand a whole tome, you must be selective about which books are a priority.
Being books not so common, the logical consequence is that even libraries are relatively rare, and great ones even more.
Literacy seems to be widespread about the Nobles of Westeros.
Almost every main character of the series is a <PERSON> of some sort, and we see that almost all of them can read and write without too much hassle, not just the most bookish ones (i.e. 130
<PERSON> and <PERSON>). The only illiterate is <PERSON>, that seems to be the exception to the norm.
To be literate you need books, in order to learn to read. This reinforce the hypothesis that at least a small collection of books (or a small library, however you want to call it) was available to every Noble House, or at least the most important ones; since fostering children to other families was a common custom, probably the smallest Houses, those that could not afford to maintain a small library themselves, solved the problem by sending them to be raised where books were available.
Summarizing, we can suppose that almost every Noble House kept at least a small library to be used mainly by their Maesters to perform their duties, and to educate their children. 738