Can I use chains on the rear wheels of my 2004 RAM 3500 when in 4WD or will that do damage to the transmission?

I dont see why not...

The stress would be to the transfer case not the trans. You will be fine.

If the conditions are bad enough to need chains on the rear, you'll have plenty of slippage and it won't matter.

You won't have any issues mechanically, but you may have some interference issues with the chains.

Make sure you put them on correctly, use side tensioners and don't go very fast with them....if they break or slip off they can take alot of stuff along with them, including fender, brake lines, etc.