The first blessing in the Book of Blessings is “Order for the Blessing of a Family”. Some rubrics for this ceremony:
“57 A minister who is a priest or deacon says the prayer of blessing with hands outstretched over the family members; a lay minister says the prayer with hands joined.”
“59 As circumstances suggest, the minister in silence may sprinkle the family with holy water.”
So it would be incorrect to say that a lay person cannot use an aspergillum.
In the General Introduction to the Book of Blessings it explains how, for example, a priest should not preside at a blessing if a bishop is present: “Priests therefore may preside at the celebration of all the blessings in this book, unless a bishop is present as presider.” Similarly a lay person would not preside if a Priest were present.
Regarding blessings by lay people it has: “Such laypersons exercise this ministry in virtue of their office (for example, parents on behalf of their children) or by reason of some special liturgical ministry …”.
[Excerpts from the English translation of Book of Blessings , © 1987, International Commission on English in the Liturgy Corporation. All rights reserved.]