There is no point in blessing a “chapel veil” as it has no sacramental significance nor are there any real blessings attached to wearing this article of clothing. it would be like blessing your favorite pair of shoes. Headcoverings were simply normal pieces of clothing worn by men and women in the apostolic era in the Middle East and elsewhere. It is not a charm, somehow shielding someone from evil.
That being said, the reason why veils of sacred women might be blessed is because it does have symbolism of the woman belonging to God alone. But there again, the veil does not protect them from anything “evil”. If anything, satan is more than eager to attack consecrated women as they are bigger “prizes”.
Edit: Also, to be clear, many sacred virgins, myself included, were given unblessed bridal veils by our bishops in the solemn ritual elevating us to the role of Bride of Christ. Believe me, if the Fathers of the Church thought it would be a good idea to have veils for lay-women blessed, they would have written about it. The Fathers had a hard enough time telling the sacred virgins to wear the headcovering (veil) of married women in public, and they would have hyped the sacramental aspect if it were a sacramental in their long discourses on the veiling of virgins. But no, the Fathers were asking the Brides of Christ to wear the ordinary attire of a married laywoman woman of the day, which meant to wear an opaque headcovering showing no hair instead of having their heads uncovered like unmarried women.