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Yes, $\Theta$ injects into $\mathcal{P}(\mathbb{R})$ (equivalently, into $2^{\mathbb{R}})$ in $L(\mathbb{R})$. The OD sets of reals are Wadge-cofinal and $\Theta$ is regular, so we can define an injection $F : \Theta \to \mathcal{P}(\mathbb{R})$ by induction: Let $F(\alpha)$ be the $<_{OD}$-least OD set of reals with Wadge rank greater the Wadge ranks of all the sets of reals $F(\gamma)$, $\gamma < \alpha$.