If $X$ is a compact Hausdorff space, we can consider the Grothendieck ring of real vector bundles on $X$,
$\mathit{KO}^0(X)$, and this extends to a generalized cohomology theory represented by a ring spectrum
$\mathit{KO}$. Using complex vector bundles, we get another generalized cohomology theory, represented by
$\mathit{KU}$. Using quaternionic vector bundles, we get a third one, represented by $\mathit{KSp} \simeq
\Sigma^4\mathit{KO}$.
In the same way, one can define the Grothendieck ring of oriented vector bundles $\mathit{KSO}(X)$, spin vector
bundles $\mathit{KSpin}(X)$, and so on for any $G$-structure. Do these
constructions extend to spectra $\mathit{KSO}$, $\mathit{KSpin}$, and so on? If not, what issues arise?
The first concern I would imagine is a failure of Bott periodicity, but $\pi_k(\mathit{SO})$ and
$\pi_k(\mathit{Spin})$ agree with $\pi_k(O)$ for $k\ge 2$, so in at least these cases some sort of
construction might be possible.
An alternative approach sidestepping Bott periodicity would be to use algebraic $K$-theory: if $\mathsf{Vect}_{k}$ denotes the topological
symmetric monoidal category of $k$-vector spaces, then $K(\mathsf{Vect}_{\mathbb R}) = \mathit{ko}$ and
$K(\mathsf{Vect}_{\mathbb C}) = \mathit{ku}$. So if $\mathsf{Vect}^{\mathrm{or}}$ denotes the topological symmetric monoidal
category of oriented real vector spaces, would it be reasonable to define $\mathit{kso}:= K(\mathsf{Vect}^{\mathrm{or}})$? Is
this an interesting object? This could also generalize to $G$-structures.
Someone must have thought about this, but I can't find it written down anywhere.
Edit: Though I've accepted Denis' excellent answer, I would still be interested to learn of existing references which consider these kinds of cohomology theories/spectra.
Best Answer
I am not sure if this is going to be a real answer to the question. However I believe these observations might be interesting.
Let me briefly sketch a way to describe a $G$-structure in (excessively) wide generality. Consider a fibration of spaces $\theta:X\to \coprod_n BO_n$. Then a $\theta$-structure on a vector bundle $V\to B$ is just a lift along $\theta$ of the map $X\to \coprod_n BO_n$ classifying $V$. Examples are orientations (with $X=\coprod_n BSO_n$), spin structures ($X=\coprod_n BSpin_n$), complex structures ($X=\coprod_n BU_n$) etc.
Now you want to define the sum of vector bundles with $\theta$-structure. In order to do so we need to require a little bit more of the map $\theta$. I'm going to assume that $X$ is an $E_\infty$-space and that $\theta$ is equipped with the structure of map of $E_\infty$-spaces. Then the abelian group of vector bundles with $\theta$-structure on some base $B$ is easily seen to be $$ \pi_0\mathrm{Map}(B,X)$$ Note that $\mathrm{Map}(B,X)$ has a canonical $E_\infty$-structure inherited from $X$. So it is natural to define the connective K-theory of vector bundles with $\theta$-structure as $$ K\theta^i(B) = \pi_{-i}\mathrm{Map}(B,X)^+$$ where $(-)^+$ is the group completion of $E_\infty$-spaces.
When the group completion of $X$ is given by a filtered colimit over the divisibility poset of $\pi_0X$ (for example $X=\coprod_n BO_n$, $X=\coprod_n BU_n$ or $X=\coprod_n BSp_n$) and $B$ is a finite space (that is the homotopy type of a CW complex with finitely many cells) you can rewrite the right hand side as $$ K\theta^i(B) = \pi_{-i}\mathrm{Map}(B,X^+)$$ since filtered colimits in spaces commute with finite limits. As Marc Hoyois notes in the comments, this is always true if every connected component of $X$ is simply connected. So $K\theta^i(-)$ restricted to finite spaces is a cohomology theory represented by the connective spectrum $X^+$.
Note that you might want a little more of this theory: you might want your $X^+$ to be an $E_\infty$-ring and not just a connective ring spectrum. This corresponds to the assumption that $X$ is an $E_\infty$-ring space and that $\theta$ respects this structure.
I do not know of a way to generalize nonconnective K-theory in this situation (of course one could just $K(1)$-localize, but that is a rather ad hoc procedure).
Also, the map $\theta$ is obviously just along for the ride, but since it allows us to think of $K\theta$ as "K-theory of vector bundles with structure" I think it helps the intuition.