In this day and age many couples still like the idea and tradition of the Giving of the Bride, but perhaps not what is stands for or the wording that goes with it.
If the bride wants someone to escort her down the aisle, she can choose instead to modify the wording to a more modern version.
Any wording can be used, but here is a sample we prepared for you as an idea:
Celebrant addresses person escorting bride (attendant)
Cel Others would ask, at this time, who gives the bride in marriage, but, as a
woman is not property to be bought and sold, given and taken, I ask simply if
she comes of her own will and if she has her family's blessing.
Celebrant: Rachel, is it true that
you come of your own free will and accord?
Rachel: Yes, it is true.
Celebrant: And with whose blessings accompany you?
Bride's attendant: She’s accompanied
with her father’s blessings.