Did/Does Islam have a form of divination or fortune telling?

Short answer: totally.

Long answer: It depends on the culture; not all cultures in which people practiced Islam had the same forms of divination.

A. Afzar Moin in the Millennial Sovereign documents all sort of occult practices amongst the Mughal elites of India, including, for example, prophetic dream visions, astrology, date-related millennialism, the reading of omens in unusual events, the correlation of names and dates to read the future, and all sorts of other interesting stuff.

Robert Finlay has a whole article, "Prophecy and Politics in Istanbul: Charles V, Sultan Süleyman, and the Habsburg Embassy of 1533-1534" that talks at length on the way in which portents, such as revolving rocks and phantom warriors in the sky arose and were acted upon by influential figures in the Ottoman empire during that time period.

Arab and Persian astronomers/astrologers were hugely influential in spreading astronomical/astrological knowledge to Europe. To quote Pankenier's "The Planetary Portent of 1524 in China and Europe," because it happens to be open in front of me for unrelated reasons, "With regard to planetary conjunction periods in particular, the most influential contribution after Ptolemy was made by a ninth-century Persian Christian astrologer... Abu Ma'shar... held to be the single most important transmitter of Aristotle's theories of nature to the Latin West" (Pankenier 2009, 365). Abu Ma'shar was not, of course, a Muslim himself, but his works were well circulated in the Muslim world. He combined Aristotle with "hellenistic horoscopy, zoroastrian millennarianism... [and the] Indian yuga of 180,000 years" to predict the future out to the year 176,899 CE (Pankenier 2009: 365).

That's just a very, very cursory run through some stuff I have hanging around, but I think it illustrates the point well enough. Islamic cultures had a huge variety of divinatory practices, some of which were extraordinarily sophisticated and influential.