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.

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?