Maps preserving common zeros between subspaces of vector-valued continuous functions

For metric spaces $X$ and $Y$, normed spaces $E$ and $F$, and certain subspaces $A(X,E)$ and $A(Y,F)$ of vector-valued continuous functions, we obtain a complete characterization of linear and bijective maps $T:A(X,E)\to A(Y,F)$ preserving common zeros, that is, maps satisfying the property \setcounter{equation}{15} \label{dub} Z(f)\cap Z(g)\neq \emptyset \Longleftrightarrow Z(Tf)\cap Z(Tg)\neq \emptyset for any $f,g\in A(X,E)$, where $Z(f)=\{x\in X:f(x)=0\}$. Moreover, we provide some examples of subspaces for which the automatic continuity of linear bijections having the property (\ref{dub}) is derived.