This is definitely not true; in fact, it's easy to construct counterexamples with $A$ finite dimensional.
For any Lie algebra $\mathfrak{g}$, you can define a Poisson algebra structure on $\mathbb{C}\oplus \mathfrak{g}$ such that $(a_1+X_1)(a_2+X_2)=a_1a_2+a_1X_2+a_2X_1$ for $a_i\in \mathbb{C}$ and $X_i\in \mathfrak{g}$ (you can more fancily define this as $\mathrm{Sym}(\mathfrak{g})/ \mathfrak{g}^2$); the Poisson bracket is defined by $\{a_1+X_1,a_2+X_2\} =[X_1,X_2]$. The simple Poisson modules over this algebra are precisely the simple modules over the Lie algebra (with the "commutative" action of $\mathfrak{g}$ trivial, and the "bracket" action being the representation). It's well known that finite dimensional Lie algebras have lots of simple infinite dimensional modules; for example, consider the Verma module for a generic highest weight over $\mathfrak{sl}_2$. These aren't finitely generated over $A$, since a finitely generated $A$ module is finite dimensional.
EDIT: I hadn't noticed the comment about reducedness in the question, but you can think of these same examples as modules over $\mathrm{Sym}(\mathfrak{g})$; more generally, as argued in this paper of David Jordan, these sort of quotients show up as $A/I^2$ whenever you have a maximal ideal $I$ which is Poisson (and every finite dimensional simple is pulled back from this construction).