Not in general.
For instance, if $A$ is simple then this is not possible. In fact, any isogeny $$f \colon E_1 \times \ldots \times E_n \to A$$ would give a dual isogeny $$f^{\vee} \colon A^{\vee} \to E_1^{\vee} \times \ldots \times E_n^{\vee},$$ and the pullback in $A^{\vee}$ of any elliptic curve in $E_1^{\vee} \times \ldots \times E_n^{\vee}$ would give elliptic curves in $A^{\vee}$. This is a contradiction: indeed $A^{\vee}$ is simple, since by assumption $A$ is simple.