  We first investigate the algebraic structure of vertex algebroids $B$ when
$B$ are simple Leibniz algebras. Next, we use these vertex algebroids $B$ to
construct indecomposable non-simple $C_2$-cofinite $\mathbb{N}$-graded vertex
algebras $\overline{V_B}$. In addition, we classify $\mathbb{N}$-graded
irreducible $\overline{V_B}$-modules and examine conformal vectors of these
$\mathbb{N}$-graded vertex algebras $\overline{V_B}$.
