The $R$-boundedness of certain families of vector-valued stochastic convolution operators with scalar-valued square integrable kernels is the key ingredient in the recent proof of stochastic maximal $L^p$-regularity, $2<p<\infty$, for certain classes of sectorial operators acting on spaces $X=L^q(μ)$, $2\le q<\infty$. This paper presents a systematic study of $R$-boundedness of such families. Our main result generalises the afore-mentioned $R$-boundedness result to a larger class of Banach lattices $X$ and relates it to the $\ell^{1}$-boundedness of an associated class of deterministic convolution operators. We also establish an intimate relationship between the $\ell^{1}$-boundedness of these operators and the boundedness of the $X$-valued maximal function. This analysis leads, quite surprisingly, to an example showing that $R$-boundedness of stochastic convolution operators fails in certain UMD Banach lattices with type $2$.