  We prove that scalar-valued sparse domination of a multilinear operator
implies vector-valued sparse domination for tuples of quasi-Banach function
spaces, for which we introduce a multilinear analogue of the UMD condition.
This condition is characterized by the boundedness of the multisublinear
Hardy-Littlewood maximal operator and goes beyond examples in which a UMD
condition is assumed on each individual space and includes e.g. iterated
Lebesgue, Lorentz, and Orlicz spaces. Our method allows us to obtain sharp
vector-valued weighted bounds directly from scalar-valued sparse domination,
without the use of a Rubio de Francia type extrapolation result.
  We apply our result to obtain new vector-valued bounds for multilinear
Calder\'on-Zygmund operators as well as recover the old ones with a new sharp
weighted bound. Moreover, in the Banach function space setting we improve upon
recent vector-valued bounds for the bilinear Hilbert transform.
