  We prove a sparse bound for the $m$-sublinear form associated to
vector-valued maximal functions of Fefferman-Stein type. As a consequence, we
show that the sparse bounds of multisublinear operators are preserved via
$\ell^r$-valued extension. This observation is in turn used to deduce
vector-valued, multilinear weighted norm inequalities for multisublinear
operators obeying sparse bounds, which are out of reach for the extrapolation
theory recently developed by Cruz-Uribe and Martell. As an example,
vector-valued multilinear weighted inequalities for bilinear Hilbert transforms
are deduced from the scalar sparse domination theorem of the authors.
