Assuming your drawers are full depth, you don't have a vertical, full length strongback in that structure. If the drawers were half-depth, with a second set of drawers on the back side (for seldom-used stuff), then you could have a vertical center spine of 3/4" ply running lengthwise down the middle, and have plenty of rigidity lengthwise.
What size angle-iron do you have under it now? I would think 2" real angle-iron (not that stamped and bent sheet metal crap) would be enough, and use two, full length, far enough in from the sides so they don't interfere with the casters' swiveling.
Otherwise, since you have ~4" casters under it, you should have room for almost 4" deep vertical beams, glued and screwed into 1/2" deep, snug dadoes running the length of the bottom. I'd use 4 or more. 3/4" thick is fine.
-- Andy - Arlington TX