The kinetic term in the Lagrangian density for a chiral superfield $A$ coupled to a vector superfield $V$ is $A^\dagger e^V A$. The correct gauge transformation of $A$ is given by $A \rightarrow e^{i \phi} A$ where $\phi$ is indeed a chiral superfield to preserve the chirality of $A$, and so with values in the complexified Lie algebra of the gauge group. Under this transformation, $A^\dagger \rightarrow A^\dagger e^{-i \phi^\dagger}$ and so

$A^\dagger e^V A \rightarrow A^\dagger e^{-i \phi^\dagger } e^{V} e^{i \phi} A$.

As the kinetic term has to be gauge invariant, this shows that the correct transformation of the vector superfield $V$ under a gauge transformation is $V \rightarrow V'$ such that $e^{V'} = e^{-i \phi^\dagger } e^{V} e^{i \phi}$ i.e. $V' = V+i(\phi - \phi^\dagger)+...$ (the ... are here if the gauge group is non-abelian because then one has to be careful about exponentials) and this is indeed the correct gauge transformation which preserves the reality condition satisfied by the gauge superfield, $V=V^\dagger$, which guarantees that the gauge group of the theory is indeed the compact gauge group and not its complexification.

Conclusion: in $\mathcal{N}=1$ $4d$ superfield formalism, gauge transformations are parametrized by chiral superfields, thus preserving chirality of the various charged matter fields, but this chiral superfield enters in the gauge transformation of the gauge field with its conjugate in a way preserving the reality condition of the gauge field.

One standard reference for this kind of questions is the book by Wess and Bagger, "Supersymmetry and supergravity".