Yes.  Assume ZFC.  If there is a proper class of inaccessible cardinals, then Tarski's Axiom A holds because whenever $\kappa$ is inaccessible, the rank initial segment $V_\kappa$ of $V$ is a Tarski set.  Conversely, if Tarski's Axiom A holds then for every set $x$ there is a Tarski set $y$ with $x \in y$, and it's not hard to show that $|y|$ is an inaccessible cardinal greater than $|x|$, so there is a proper class of inaccessible cardinals.