It depends on what version of connection on principal 2-bundles you consider. There are at least four versions:
Fake-flat connections, these are the ones that have a well-defined 2-dimensional parallel transport
Regular connections
Adjusted connections (this requires additional structure on your Lie 2-group)
General connections
Versions 1, 2, and 4 are discussed in my paper "A global perspective to connections on principal 2-bundles". Version 3 is discussed, e.g., in "Explicit Non-Abelian Gerbes with Connections" by Rist, Saemann, and Wolf.
EDIT: the underlying structure in all four versions is a Lie 2-algebra-valued 1-form $\Omega$ on the total space of the principal 2-bundle (which is a Lie groupoid). Note that a 1-form involves (because a Lie 2-algebra is considered to live in degrees $-1$ and $0$) two 1-forms and one 2-form. The 1-form $\Omega$ satisfies a condition completely analogous to the condition imposed on connection 1-forms on ordinary principal bundles. Without further conditions, this is version 4. The other versions impose conditions on the curvature 2-form.
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.
For the String-2-group, every String-2-bundle admits a string connection in the sense developed in my paper "String Connections and Chern-Simons Theory", and the space of such string connections is contractible. However, as far as I know, it is not worked out to which of the above versions of connections on principal 2-bundles these string connections correspond. Probably, it is version 3.