  We revisit the construction of integral forms for vertex (operator) algebras
$V_L$ based on even lattices $L$ using generators instead of bases, and we
construct integral forms for $V_L$-modules. We construct integral forms for
vertex (operator) algebras based on highest-weight modules for affine Lie
algebras and we exhibit natural generating sets. For vertex operator algebras
in general, we give conditions showing when an integral form contains the
standard conformal vector generating the Virasoro algebra. Finally, we study
integral forms in contragredients of modules for vertex algebras.
