Does the Eastern Orthodox Church eat leavened or unleavened bread during the Eucharist, and if it is indeed leavened bread, what is their basis for this practice?
Yes. The Eastern Orthodox Church uses leavened bread.
Leavened bread has always been used in the Eastern Church. In fact, at one point in time, a great controversy raged over the fact that in the Eastern Church leavened bread was used, while in the Christian West unleavened bread was the norm. (Orthodox Church in America)
The Orthodox Church uses leavened bread for, according to the Gospel of Saint John, Last Supper and Passion, took place during the evening, night and day time of Passover Day, therefore leavened bread was eaten in Last Supper. (Orthodox Wiki)
The words used to describe the bread of the Last Supper (and Christ's commandment to do as he had done) are the basis for this.
If we look at the word for the bread used at the Last Supper is artos, the Greek generic term for bread that almost always...though not necessarily always referred to every day leavened break. The Greek term that specifically means unleavened bread is azymes. This gap is cleared up if we look at the earliest Aramaic translations of the Gospels (third century or earlier), the word they use to translate bread at the Last Supper is not generic, but specifically means leavened bread. (Orthodox Christianity, emphases mine)