The statement is true for Lie groups and principal bundles, with every principal bundle admitting a connection and I see no reason for the analogue result not to hold in the Lie $2$-group case but I can't find the precise statement anywhere. As a bonus question, given a Lie $n$-group $G$, do principal $G$ $n$-bundles always admit an $n$-connection?

What I am mainly concerned about is the case of the Lie $2$-group $G=String(k)$ (bonus: the other $n$-groups appearing in the Whitehead tower of the orthogonal group).