My wife is dressing my son Francis up as... St. Francis for an All Saints party. I think the timeline is thus that St. Francis could have prayed the Rosary, but it was more of a Dominican thing to do. Would he have prayed the Rosary or would he have avoided it as being something the distinguished his order from St. Dominic's?
It is doubtful that St. Francis would have prayed what we know now as the contemporary rosary consisting of five decades of Hail Marys. The two saints were contemporaries as St. Dominick lived AD 1170- AD 1221 and St. Francis lived AD 1181- AD 1226. St. Dominick was a Spanish Priest and founder of the Dominican orders. St. Francis of Assisi was the son of a very wealthy Italian merchant who heard a call to "Rebuild My Church" and lived his life in poverty. He enjoyed preaching in the countryside, avoiding the parish churches and preached the Gospel in common Latin rather than ecclesiastical or church Latin. (REF: "Roaming Catholics: ending the wandering to embrace the wonder" by Kenneth Behr, Xpyria Press 2015 page 155)
Based on historical research, the development of the practice of using beads in prayer has a very significant connection to the practice of the early monks in their Liturgy of the Hours that included reading through the 150 Psalms. This practice was abbreviated into the Breviary (from the Latin 'brevis' meaning short or concise) and associated with knots in a cord and ultimately prayer beads. The 150 knots or beads were reduced to multiple decades of tens. For the common people that could neither speak nor read Latin, a poor man's Breviary was introduced to keep track of their chants, and the difficult-to-memorize Psalms were changed to that of the Paternoster (Our Father).
The modern day rosary was most likely introduced in the 14th century by groups of Christians who had a deep devotion to the Virgin and modified the poor man's Breviary into five decades of Hail Marys rather than the Our Fathers. (REF: "Roaming Catholics, page 82)
The association of St. Dominick, the rosary and the Virgin Mary, while retold often has really no historically verifiable basis. The Catholic Encyclopedia, New Advent comes to a similar conclusion (REF: H. Thurston and A. Shipman, the Rosary, in The Catholic Encyclopedia (New York: Robert Appleton Co., 1912), New Advent http://www.newadvent.org/cathen/13184b.htm.)
"Since the Rosary is composed, principally and in substance, of the prayer of Christ and the Angelic Salutation, that is, the Our Father and the Hail Mary, it was without doubt the first prayer and the principal devotion of the faithful and has been in use all through the centuries, from the time of the apostles and disciples down to the present. It was only in the year 1214, however, that the Church received the Rosary in its present form and according to the method we use today." - http://www.theholyrosary.org/rosaryhistory
This website, along with the fact that St. Francis died in 1226, indicate that yes, he would have probably been familiar with the Rosary. Additionally, the website confirms that not only would he have been familiar with the start (or roots of) the rosary, but he would have even been familiar with the Rosary "in its present from and... the method we use today."