In this article, Fefferman-Stein inequalities in $L^p(\mathbb R^d;\ell^q)$ with bounds independent of the dimension $d$ are proved, for all $1 < p, q < + \infty.$ This result generalizes in a vector-valued setting the famous one by Stein for the standard Hardy-Littlewood maximal operator. We then extend our result by replacing $\ell^q$ with an arbitrary UMD Banach lattice. Finally, we prove similar dimensionless inequalities in the setting of the Grushin operators.