How thick should a plate steel stringer be?
Given that your stringers are going to be about 8" wide, you'll be fine with some pretty thin stuff. Heavy Jeep frames from the 1970s were only about 3/16" thick, and only about 4" wide. I've seen truck frames around 1/4" by 8", but they were larger single-axle agricultural trucks. Really large single-axle trucks only have about 1/2"-thick steel frames.

Come to think of it, I've walked stairs made with no heavier than 3/16" stringers in penitentiaries. They felt nearly like STONE.

If you're concerned about racking, remember that every tread will act as an anti-rack brace. Until your treads go on, the stringers will feel pretty flimsy if you push them sideways. AFTER the treads go on, they'll be train-wreck rigid.