  We consider the notion of the graph product of actions of groups
$\left\{G_v\right\}$ on a $C^*$-algebra $\mathcal{A}$ and show that under
suitable commutativity conditions the graph product action $\bigstar_\Gamma
\alpha_v: \bigstar_\Gamma G_v \curvearrowright \mathcal{A}$ has the Haagerup
property if each action $\alpha_v: G_v \curvearrowright \mathcal{A}$ possesses
the Haagerup property. This generalizes the known results on graph products of
groups with the Haagerup property. To accomplish this, we introduce the graph
product of multipliers associated to the actions and show that the graph
product of positive definite multipliers is positive definite. These results
have impacts on left transformation groupoids and give an alternative proof of
a known result for coarse embeddability. We also record a cohomological
characterization of the Haagerup property for group actions.
