Vick rose to stardom with the Atlanta Falcons before his involvement in a dogfighting ring led to him spending 21 months in federal prison.
The 37-year-old spent six seasons with the Falcons and displayed superb running ability with a strong but often erratic arm. He threw for 71 touchdowns, rushed for 21, and was a three-time Pro Bowl selection before the dog-fighting conviction.
He passed for 22,464 yards and 133 touchdowns against 88 interceptions during 13 NFL seasons. He added 6,109 yards and 36 rushing touchdowns.
Childress, 61, left Kansas City at the end of last season and joined former Chiefs offensive coordinator Matt Nagy in Chicago, helping with his transition to head coach of the Bears.
Atlanta will play its home games at Georgia State Stadium, which was known as Turner Field.
Atlanta is the second city of eight for the new league, which announced earlier this month that Steve Spurrier would coach the team in Orlando, Fla.