Books in general were more common in the pre-printing press Islamic world for the simple fact that after the 8th century the Islamic world had access to paper. The story goes that paper-makers were captured in the battle of Talas in 751 when an Islamic army defeated a Song-dynasty Chinese army but it's the kind of story that sounds a bit too good to be true. In any event, paper moved west into Islamdom at about that time. Paper was introduced to Christian Europe via Islamic Spain around the 12th century. Paper has the advantage over, say, vellum of being cheap, easy to produce and easy to write on.

While the Ottomans used printing presses much earlier there was no dedicated printing press in the Arab world until the end of the 18th century when Napoleon brought a printing press to Egypt.

Between those two dates I'm not really familiar with a way to provide a quantitative answer beyond "widespread." Anyone who wanted a quran and could read one would almost certainly have been capable of finding and purchasing one, at least in major urban centers. Booksellers were common and even rarer texts, let alone something as ubiquitous as the Quran, appear to have been readily available. Ibn Sina's autobiography describes him basically stumbling across a copy of Al Farabi's commentary on metaphysics at a bookstall and which he purchases for a relatively small amount of money.

Even in small villages however, at least one male member of each family was expected to have had some basic attendance at a maktab school, and all were expected to have been versed in the quran well enough to perform prayer. Each of these schools would have taught some form of Arabic literacy (with an emphasis on rote memorization) by someone who had memorized the quran. That issue of memorization is important because unlike in the Christian tradition there was a much wider expectation that individual Muslims have at least some portion, if not the entirety, of the Quran memorized.