= Solution
Use the mostly-plus metric, so $\Box$ reduces to $\nabla^2$ for static fields. Seek a <Green function> satisfying $(\nabla^2-\lambda^2)G=\delta^{(3)}$. Convolving it with $-\mu_0\mathbf J$ gives the required solution.
For $R>0$, write $G=f(R)/R$. The homogeneous equation is $f''-\lambda^2f=0$. Decay discards the growing exponential, while the normalization at the origin follows from $\nabla^2(1/R)=-4\pi\delta^{(3)}$. Therefore \b[for every $\lambda\geq0$],
$$
\boxed{G(R)=-\frac{e^{-\lambda R}}{4\pi R},\qquad \mathbf A(\mathbf x)=\frac{\mu_0}{4\pi}\int\frac{e^{-\lambda|\mathbf x-\mathbf x'|}}{|\mathbf x-\mathbf x'|}\mathbf J(\mathbf x')\,d^3x'.}
$$
The $1/R$ singularity fixes the delta coefficient; the extra factor is smooth to leading order at zero. For a conserved localized static current, integration by parts also gives $\nabla\cdot\mathbf A=0$, as required by the constraint.
At $\lambda=0$ the far field has algebraic multipole decay; for an ordinary localized steady current the vector potential starts at dipole order. At $\lambda>0$ the <Yukawa potential> gives exponential suppression on length scale $\lambda^{-1}$, with an overall bound of order $e^{-\lambda|\mathbf x|}/|\mathbf x|$ for bounded source support.
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