I'm not sure how to specify it, I mean somehow that you shouldn't pass to topological spaces with some realization. So, for example just taking the geometric realization of a simplicial set and developing K-theory for the realization would not be "internal" to the category.
For vector bundles over a topological space X, the data required to specify it doesn't draw from any other categories than topological spaces and vector spaces if I recall correctly. We can then form the grothendieck group on vector bundles and get a version of k-theory. So can something analogous work for kan complexes …