Assume $M_1$ and $M_2$ are two contractible compact manifolds with boundary and that $\partial M_1 = \partial M_2 = \Omega$. Fix a homeomorphism between the boundaries and consider $X = M_1 \sharp M_2$ the space obtained by gluing those two manifolds by their boundaries. The example I have in mind is the sphere $S^2$ given by union of two closed disks glued by their boundaries.
Let $VO(X)$ be the abelian semigroup given by all real vector bundles over $X$ with fiberwise direct sum and $VU(X)$ the corresponding semigroup for complex vector bundles.
Question 1: Can the structure of $VO(X)$, resp. $VU(X)$, be explicitly described?
Let $E \to X$ is one such bundle, assume it is $n$-dimensional. $E$ restricted to the interior of $M_1$ is trivial and the same holds for the restriction to $M_2$. The gluing of the fibers over $\partial M_1$ will be given by a map $\Omega \to O(n)$, resp. $\Omega \to U(n)$. It seems intuitive that homotopy-equivalent maps induce equivalent vector bundles and thus we have maps $$ [\Omega, O(n)] \to VO(X) \quad \mbox{and} \quad [\Omega, U(n)] \to VU(X) $$ (Here [X, Y] denotes the space of cont. maps $X \to Y$ up to homotopy)
If we define the abelian semigroups $WO(\Omega)$ and $WU(\Omega)$ by taking $$ WO(\Omega) = \bigcup_{n \geq 0} [\Omega, O(n)] \quad \mbox{and} \quad WU(\Omega) = \bigcup_{n \geq 0} [\Omega, U(n)]. $$ with addition given by direct sums, i.e. for $\phi_1: \Omega \to O(n_1)$ and $\phi_2: \Omega \to O(n_2)$ we have $\phi_1 + \phi_2:= \phi_1 \oplus \phi_2: \Omega \to O(n_1) \oplus O(n_2) \subset O(n_1 + n_2)$
Question2: Does the natural map induce homomorphisms of abelian semigroups $$ WO(\Omega) \to VO(X) \quad \mbox{and} \quad WU(\Omega) \to VU(X). $$ If so, are those maps surjective?
I will rule out the possibility of those maps being injective, at least in the real case, since $O(1) = \{+1,-1\}$ but the only real $1$-dimensional bundle over the sphere $S^2$ is the trivial one.
(I am a newcomer to K-theory, just trying to grasp a few basic concepts. Pointers are welcome)
For the first question, you have almost defined the $K$-theory of a group, the Grothendieck completion of $VO(X)$ is $K(X)$, the Chern character induces an isomorphism between $K(X)\otimes\mathbb{Q}\rightarrow H^{2*}(X,\mathbb{Q})$.
https://en.wikipedia.org/wiki/Topological_K-theory