The direct sum of real vector bundles endows $BO=\mathrm{colim} BO(n)$ with a natural structure of abelian group up to homotopy. The same applies to the classifying spaces of all groups in the Whitehead tower of $O$, i.e., one has a natural structure of abelian group up to homotopy on $BSO$, $BSpin$, $BString$, etc.


As $w(E\oplus F)=w(E)\cup w(F)$ one sees that 
$$
w\_1\colon BO \to K(\mathbb{Z}/2\mathbb{Z},1)
$$ 
is a morphism of abelian groups up to homotopy, and that similarly
$$
w\_2\colon BSO \to K(\mathbb{Z}/2\mathbb{Z},2)
$$ 
is a morphism of abelian groups up to homotopy. One can even make a step further and see
$$
BSO \to BO\xrightarrow{w\_1} K(\mathbb{Z}/2\mathbb{Z},1)
$$
and 
$$
BSpin \to BSO\xrightarrow{w\_2} K(\mathbb{Z}/2\mathbb{Z},2)
$$
as ``short exact sequences of abelian groups up to homotopy''. 


One can be more ambitious here. Not only $BO$ is an abelian group up to homotopy, but it is an $\infty$-loop space, i.e. $BO=\Omega^\infty bo$ for a certain connective spectrum $bo$. The same applies to $BSO$, $BSpin$,etc., and it also applies to $K(\mathbb{Z}/2\mathbb{Z},n)$ as $K(\mathbb{Z}/2\mathbb{Z},n)=\Omega^\infty\Sigma^n H\mathbb{Z}/2$. So one may hope that the above sequences are actually infinitely deloopable and come from fibrations
$$
bso \to bo\xrightarrow{\Omega^{-\infty}w\_1} \Sigma H\mathbb{Z}/2
$$
and 
$$
bspin \to bso\xrightarrow{\Omega^{-\infty}w\_2} \Sigma^2 H\mathbb{Z}/2
$$
of connective spectra. Versions of this latter statement seem to appear in the literature, at least in the form "$BSO \to BO\xrightarrow{w\_1} K(\mathbb{Z}/2\mathbb{Z},1)$ is a fibration of infinite loop spaces" which however I am only able to give a precise meaning by interpreting it as above. For instance one finds: "Recall that $BSpin^c$ participates in a fibration of infinite loop spaces $K(\mathbb{Z},2)\to BSpin^c\to BSO\xrightarrow{bw\_2}K(\mathbb{Z},3)$'' in section 7 of Ando-Blumberg-Gepner's [Twists of K-theory and TMF](https://arxiv.org/pdf/1002.3004.pdf). 


My question is: 


* Is it true that the above are indeed fibrations of connective spectra inducing the usual fibrations of topological spaces via $\Omega^\infty$?
* Where can I find a rigorous proof of this statement?



How are characteristic classes morphisms of infinite loop spaces? (if they are)
Your sequences are all arise in the following standard way. Suppose $x$ is an $(n-1)$--connected spectrum and let $X = \Omega^{\infty} x$. One always has a 
fibration sequence 
$$y \rightarrow x \rightarrow \Sigma^n H\pi\_n(X)$$
and applying $\Omega^\infty$ to this yields a fibration sequence of spaces
$$Y \rightarrow X \rightarrow K(\pi\_n(X),n).$$


$Y$ is the $n$--connected cover of $X$. 


(In your situation, one has successive covers $bspin \rightarrow bso \rightarrow bo$.)


