Why not $\mathit{KSO}$, $\mathit{KSpin}$, etc.?

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.

Regarding preexisting references, I found a few examples of people using Grothendieck groups of oriented or spin vector bundles, but no cohomology theories or spectra. I'll leave them here in case anyone else is interested.

For $\mathit{KSpin}$:

• Li-Duan, “Spin characteristic classes and reduced $\mathit{KSpin}$ group of a low-dimensional complex” looks at the reduced Grothendieck group $\widetilde{\mathit{KSpin}}(X)$.
• Sati, “An approach to M-theory via KSpin” applies Li-Duan's construction to string theory.
• Karoubi, “Clifford modules and invariants of quadratic forms” defines a group $\mathit{KSpin}(X)$ in a different way.

For $\mathit{KSO}$:

• Pejsachowicz, “Bifurcation of Fredholm maps II: the dimension of the set of bifurcation points.”