I did that exact swap from a 5-speed to the 32RH in my '97 a few months ago. Yes you need the automatic style transmission mounting bracket and the mount itself. You'll also need a blind bearing puller to pull the pilot bearing and its retainer sleeve out of the end of the crankshaft, and you'll need the automatic style flywheel as well.
The engine computer needs to be swapped for one out of another TJ of the same year as yours with the same engine but for an automatic transmission which is what is needed to lock up the automatic's lockup torque converter. The wiring is already there for the lockup torque converter, hanging above the transmission location, it just needs to be plugged in.
You'll need to replace the new automatic transmission's output shaft seal with an extended output shaft seal available from AA, Tera, etc. that will seal around the transfer case's input shaft as your transfer case input shaft's dimensions are meant to match up with a 5-speed. The transmission will leak if that seal is not there or if it's damaged during installation since it's a rather hard seal to get seated properly. Remember to grease the rubber part of the new seal.
You'll likely need new hard lines for the cooler as they change how they're bent between some model years. Hopefully your TJ is 2002 or older since I can't imagine the electronics in a 2003 being happy with that transmission that was only used until 2002.