  In this paper, first we introduce the notion of a $\VB$-Lie $2$-algebroid,
which can be viewed as the categorification of a $\VB$-Lie algebroid. The
tangent prolongation of a Lie $2$-algebroid is a $\VB$-Lie $2$-algebroid
naturally. We show that after choosing a splitting, there is a one-to-one
correspondence between $\VB$-Lie $2$-algebroids and flat superconnections of a
Lie 2-algebroid on a 3-term complex of vector bundles. Then we introduce the
notion of a $\VB$-$\LWX$ 2-algebroid, which can be viewed as the
categorification of a $\VB$-Courant algebroid. We show that there is a
one-to-one correspondence between split Lie 3-algebroids and split $\VB$-$\LWX$
2-algebroids. The notion of a $\VB$-Lie $2$-bialgebroid is introduced and the
double of a $\VB$-Lie $2$-bialgebroid is a $\VB$-$\LWX$ 2-algebroid. Finally,
we introduce the notion of an $E$-$\LWX$ 2-algebroid and show that associated
to a $\VB$-$\LWX$ 2-algebroid, there is an $E$-$\LWX$ 2-algebroid structure on
the graded fat bundle naturally. By this result, we give a construction of a
Lie 3-algebra from a given Lie 3-algebra, which provides interesting examples
of Lie 3-algebras including the higher analogue of the string Lie 2-algebra.
