  This is a sequel to \cite{li-qva1} and \cite{li-qva2} in a series to study
vertex algebra-like structures arising from various algebras such as quantum
affine algebras and Yangians. In this paper, we study two versions of the
double Yangian $DY_{\hbar}(sl_{2})$, denoted by $DY_{q}(sl_{2})$ and
$DY_{q}^{\infty}(sl_{2})$ with $q$ a nonzero complex number. For each nonzero
complex number $q$, we construct a quantum vertex algebra $V_{q}$ and prove
that every $DY_{q}(sl_{2})$-module is naturally a $V_{q}$-module. We also show
that $DY_{q}^{\infty}(sl_{2})$-modules are what we call
$V_{q}$-modules-at-infinity. To achieve this goal, we study what we call
$\S$-local subsets and quasi-local subsets of $\Hom (W,W((x^{-1})))$ for any
vector space $W$, and we prove that any $\S$-local subset generates a (weak)
quantum vertex algebra and that any quasi-local subset generates a vertex
algebra with $W$ as a (left) quasi module-at-infinity. Using this result we
associate the Lie algebra of pseudo-differential operators on the circle with
vertex algebras in terms of quasi modules-at-infinity.
