  In this paper, we first introduce some new kinds of weighted amalgam spaces.
Then we deal with the vector-valued intrinsic square functions, which are given
by \begin{equation*} \mathcal S_\gamma(\vec{f})(x) = \Bigg(\sum_{j=1}^\infty
\big|\mathcal S_\gamma(f_j)(x) \big|^2\Bigg)^{1/2}, \end{equation*} where
$0<\gamma\leq1$ and \begin{equation*} \mathcal S_\gamma (f_j)(x) =
\left(\iint_{\Gamma(x)} \Big[\sup_{\varphi\in{\mathcal C}_\gamma}
\big|\varphi_t*f_j(y) \big|\Big]^2 \frac{dydt}{t^{n+1}}\right)^{1/2}, \quad
j=1,2,\dots. \end{equation*} In his fundamental work, Wilson established
strong-type and weak-type estimates for vector-valued intrinsic square
functions on weighted Lebesgue spaces. The goal of this paper is to extend his
results to these weighted amalgam spaces. Moreover, we define vector-valued
analogues of commutators with $BMO(\mathbb R^n)$ functions, and obtain the
mapping properties of vector-valued commutators on the weighted amalgam spaces
as well. In the endpoint case, we also establish the weighted weak $L\log
L$-type estimates for vector-valued commutators in the setting of weighted
Lebesgue spaces.
