Yes AutoZone will replace your battery for free as long as you buy the battery from them.
You can go into AutoZone and purchase a new battery and then the AutoZone Employee will come out and disconnect your old battery and then put the new battery in the old batteries place.
Then they will connect the battery all at no additional charge.
However if the battery has things such as a brace or something over the top of the battery making the battery difficult to replace then AutoZone will not replace the battery even if you buy a battery from them.
AutoZone will only replace easily accessible batteries.
If they cannot replace your car battery then they will refer you to a shop for someone to do it for you.