It depends on what version of connection on principal 2-bundles you consider. There are at least four versions:

 1. Fake-flat connections, these are the ones that have a well-defined 2-dimensional parallel transport 

 2. Regular connections
 
 3. Adjusted connections (this requires additional structure on your Lie 2-group)

 4. General connections

Versions 1, 2, and 4 are discussed in my paper "[A global perspective to connections on principal 2-bundles][1]". Version 3 is discussed, e.g., in "[Explicit Non-Abelian Gerbes with Connections][2]" by Rist, Saemann, and Wolf.

For version 1 it is clear that one cannot expect the existence of connections, since there is a quadratic equation involved. The space of fake-flat connections on a trivial bundle is not contractible.

For version 2 the existence of connections is not clear to me. 

For version 3 it seems that the question has not yet been investigated, but it could be true that every principal 2-bundle admits adjusted connections. At least this is true in certain examples of adjusted 2-groups.

For version 4, there is an existence theorem with a mild additional assumption (Theorem 5.2.14) in my above-mentioned paper. A particular weird problem in this context is that connections on 2-bundles cannot easily be pulled back along morphisms of 2-bundles. The morphism itself has first to be equipped with a kind of connection (called a "pullback" in my paper), and the existence thereof is also obstructed. 

Better results are of course possible when restricting to 2-groups of the form $BA$, where $A$ is an ordinary abelian Lie group. Then, connections of versions 1 and 2 coincide, form a contractible space, and always exist. 



  [1]: https://arxiv.org/abs/1608.00401
  [2]: https://arxiv.org/abs/2203.00092