Would it be inappropriate to wear a chapel veil while Eucharistic ministering?
First, you are not a Eucharistic Minister. You are an Extraordinary Minister of Communion.
I cannot see any problem with a woman wearing a chapel veil, a mantilla, or other head covering while helping to distribute Communion.
The more important question is whether or not help from additional lay distributors is really necessary.
We should keep in mind what Redemptionis Sacramentum says:
157 If there is usually present a sufficient number of sacred ministers for the distribution of Holy Communion, extraordinary ministers of Holy Communion may not be appointed. Indeed, in such circumstances, those who may have already been appointed to this ministry should not exercise it. The practice of those priests is reprobated who, even though present at the celebration, abstain from distributing Communion and hand this function over to laypersons.
158 The extraordinary minister of Holy Communion may administer Communion only when the priest and deacon are lacking, when the priest is prevented by weakness or advanced age or some other genuine reason, or when the number of faithful coming to Communion is so great that the very celebration of Mass would be unduly prolonged. … A brief prolongation, considering the circumstances and culture of the place, is not at all a sufficient reason.
Finally, while it is not obligatory for women to wear a head covering during Mass, I think it is a good custom (rooted in Holy Scripture) which ought to be fostered.