Dear Aaron, Grothendieck proved that there is an analytification functor $X \mapsto X^{an}$ from schemes locally of finite type over $\mathbb C$ to the category of (non-reduced!) analytic spaces, which is fully faithful when restricted to proper schemes. This induces isomorphisms from $K-$ groups in the algebraic sense on $X$ to $K-$ groups in the holomorphic sense on $X^{an}$ . This is just a mild generalization of Serre's GAGA principle proved for reduced, projective varieties. So this settles your problem in the compact algebraizable case, by telling algebraic geometers to solve it ( and they actually know quite a lot of the K-theory of schemes !)
In the diametrically opposed case of Stein manifolds, a landmark theorem of Grauert also answers your request. Namely, given a complex manifold there is an obvious forgetful functor $Vecthol(X) \to Vecttop(X)$ from isomorphism classes of holomorphic vector bundles on $X$ to isomorphism classes of topological vector bundles on the underlying topological space $X^{top}$. If $X$ is Stein, Grauert proved that the functor is an equivalence of categories : every topological vector bundle has a unique holomorphic structure. ( Results of this nature fit into what is called the "Oka principle". ) There are no extension problems for short exact sequences $0 \to \mathcal E \to \mathcal F \to \mathcal G \to 0$ because they all split: in the Stein case thanks to theorem B and in the topological case because of partitions of unity (theorem B in disguise, actually: fine sheaves are acyclic). So in the Stein case too you can relax and ask topologists to do your work .
Finally, there are complex manifolds between these extreme cases. I am not aware of a general theory there ( of course that proves nothing but my ignorance) . This looks like an interesting topic of investigation, especially in view of Winkelmann's theorem ( link to survey here) that on every compact holomorphic manifold of positive dimension $n $ there exists a non-trivial holomorphic vector bundle of rank $\leq n$.