All accumulation points of $J_p$ in $\mathbb{C}_p$ are roots of degree two monic equations over $\mathbb{Z}_p$, and their approximants are necessarily supersingular. Moreover, there exist accumulation points. (There is then a further restriction: the reduction of the point has to be one of the $\approx p/12$ supersingular residues in $\mathbb{F}_{p^2}$. Is this perhaps the only restriction?)

First, extending Pete Clark's remarks, the ordinary CM points have no accumulation point in $\mathbb{C}_p$. (So no, the CM invariants aren't dense in any of the ordinary disks.) This is similar to the corresponding fact about the $p$-adic roots of unity. The analogy here is substantiated by the Serre-Tate theory; cf. Prop. 3.5 in de Jong and Noot's paper *Jacobians with complex multiplication*. Building upon this, P. Habegger (*The Tate-Voloch conjecture in a power of a modular curve*, Int. Math. Res. Notices 2014) established much more generally that no algebraic subvariety $V/\mathbb{C}_p$ in a power of the modular curve is $p$-adically approximated by ordinary CM points not lying in $V$. The prototypical $\mathbb{G}_m^r$ case, where the special points are the torsion ones, had been established by Tate and Voloch in the same journal (*Linear forms in $p$-adic roots of unity*, 1996).

So the question reduces to approximating with supersingular points. These belong to the valuation ring of a quadratic extension of $\mathbb{Q}_p$, hence the claim in my opening paragraph. Habegger's Proposition 2 proves that $0$ is an accumulation point of supersingular CM points, in order to demonstrate that the restriction to ordinary points in his main result is essential. This should work on other examples, though I am unsure exactly which quadratic integral elements over $\mathbb{Z}_p$ may be approximated with Habegger's method. At least this shows the existence of accumulation points.