If we work over a field $k$, and take the recursive definition of $n$-vector spaces (as, e.g. in Topological Quantum Field Theories from Compact Lie Groups, arXiv:0905.0731) then a $2$-vector space is a $k$-algebra $A$, to be thought as a placeholder for its categoty of modules; morphisms between $A$ and $B$ are $B$-$A$-bimodules, ad 2-morphisms are morphisms of bimodules.

Now consider a 2-vector bundle over some space $X$. How does one see that its global sections are a 2-vector space?

The answer somehow depends on the notion of section one adopt, but in any case the relation between the various definitions should be investigated. Basically there are two notions coming to my mind:

i) natural transformations from the trivial 2-bundle to the given bundle. This is a very neat object, but it is not clear (to me) that this is a 2-vector space: which is the underlying algebra?

ii) the limit in 2-Vect of the functor from the nerve of a good open cover of $X$ to 2Vect defining the 2-bundle. This is manifestly a 2-vector space, but it is not clear (to me) that this limit exists.

Clearly the dream statement here would be that i) has a natural structure of 2-vector space, and that this 2-vector space represents the limit ii), but I'm unable to prove this.

(or the dual version of the above, under suitably finiteness assumptions)