  We compute the integral Picard group of the stack $\mathcal{M}_{2l}$ of
polarized K3 surfaces with at most rational double points of degree $2l=4,6,8$.
We show that in this range the integral Picard group is torsion-free and that a
basis is given by certain elliptic Noether-Lefschetz divisors together with the
Hodge line bundle. To achieve this result, we investigate certain stacks of
complete intersections and their Picard groups by means of equivariant
geometry. In the end we compute an expression of the class of some
Noether-Lefschetz divisors, restricted to an open substack of
$\mathcal{M}_{2l}$, in terms of the basis mentioned above.
