Steel prices are down right now. I just helped a buddy weld up a Lotus 7 frame and we used 1x1 .065 rectangular tube that I purchased for $.47 a foot locally. We built the suspension bits (a arms) from 1"x.065 round DOM which we paid $2 a foot for, but as high as that sounds it's actually pretty cheap for seamless DOM. Locally the square is sold in 24' lengths and the DOM in 20' lengths. It seems that I see a lot of karts built from really thick wall tubing. Usually around the order of .120 from what I've seen which seems like overkill to me. The kart I've been working on is made from some really thick stuff too. It looks to me like they could make these things a lot lighter if they triangulated things a bit and used thinner tubes. The thick stuff is however easier to bend without ovalizing, so maybe it's favored because it's easier to bend and weld.
FWIW I found your reference to "thin wall" and "thick wall" is a bit confusing. Those terms are pretty subjective and could mean a lot of different things. I know people who consider .120 thin, and I, and others I would imagine would consider that to be pretty thick tubing to use for frame construction.
For those who don't have good steel distributors locally I recommend onlinemetals.com and their competitor metalsdepot.com. I've used them both when ordering odd sizes in small amounts and their prices are decent.