The flag depicted on the website has no historical basis of whcih I am aware. They would have had a First National flag early on, but after being transferred to the Army of Central KY at Bowling Green in late 1861, they would have been issued the first of the Buckner/Hardee pattern battleflags - blue field with a white circle in the center and possibly the unit's name in the circle (several other flags from this issue had that).
As part of St. John Lidell's Brigade they were part of Gen. Patrick Clebrune's Division which fought the entire war under flags of this type. In fact, there are two existing flags of this type for the 5th Arkansas, both in private hands. One is of a letr issue than the January, 1862 flags and has the unit ID in the white circle while the other is for the combined 5th/13th Arkansas and this flag was issued in March, 1864. The latter was captured at the Battle of Franklin.