Yes. $0_F$ is an element of the field, just a particular instance of $a$, and this axiom guarantees that it too has an additive inverse. Things that are true for all elements of a set are true for any particular element you consider.
(I am assuming that there is a previous axiom says that $0_F$, otherwise I don't know how you could refer to it within the Axiom A.)
BTW, one could imagine a world where inverses are defined (e.g. "The inverse of an element $x$ is an element $y$ such that $x+y=0_f"$.) but we haven't yet given an axiom stating that all elements have inverses. Then we would already know that $0_F$ has an inverse, with out Axiom A, as $0_f + 0_F = 0_F$ from the definition of $0_F$.