The AMS has spent a lot of money and effort over many years to develop a fine tool that easily allows for the investigation of such questions: it is called Math Reviews (MathSciNet). The oldest mention of SW classes in the reviews seems to be a review by Whitney of a 1947 paper by Wu. There is a fantastic review (in German) by Hirzebruch of Milnor and Stasheff's book, giving some history - e.g. the SW classes go back to the mid 1930's. I believe Milnor and Stasheff themselves discuss how to think about SW classes as obstructions, and I am pretty sure that this finding this interpretation was quite close to the discovery of the formula for the SW classes via Steenrod squares: this type of question is how the `reduced squares' were discovered in the first place.
Added later: ... see Thom, René
Espaces fibrés en sphères et carrés de Steenrod. (French)
Ann. Sci. Ecole Norm. Sup. (3) 69, (1952). 109–182.
In particular, section 3 of chapter 2 carefully develops the formula for the SW classes via Steenrod squares, and compares it to earlier constructions. Chapter 1 discusses what folks might now call the Thom isomorphism (though the Gysin sequence is already known).
Remark: the basic theory of fiber bundles and classifying spaces is just being figured out at this same time.