I am not aware of any documentation allowing lay people to distribute ashes during Mass.
The Roman Missal has “The priest then places ashes on those who come forward, saying to each: …” (Roman Missal, Catholic Book Publishing Co., New York, 1985, page 77)
The Ceremonial of Bishops has:
“257 After the blessing, the appointed minister, a concelebrant or a deacon, places ashes on the bishop, as the bishop bows before him, and says, Turn away from sin and be faithful to the Gospel or Remember, man, you are dust and to dust you will return.
258 Then the bishop again puts on the mitre and, seated at the chair or standing, places ashes on the concelebrants, the ministers, and the faithful. The bishop may be assisted if necessary by some of the concelebrants or deacons.”
Monsignor Peter Elliott wrote: “If there is no other priest or a deacon present, the celebrant imposes the ashes on himself.” (Peter Elliott, Ceremonies of the Liturgical Year, Ignatius Press, 2002, ISBN 089870829X, page 57).
No mention of lay people.
In the USA the Book of Blessings has a ceremony for distributing ashes outside of Mass. It has only been approved for the USA and is not in the Latin edition of this book. It has:
“1658 If already blessed ashes are brought to the sick, the blessing is omitted and the distribution takes place immediately after the homily. The homily should conclude by inviting the sick person to prepare himself or herself for the reception of the ashes.
1659 This rite may be celebrated by a priest or deacon who may be assisted by lay ministers in the distribution of the ashes. The blessing of the ashes, however, is reserved to a priest or deacon.” (Book of Blessings, Liturgical Press, Minnesota, 1989, ISBN 0-8146-1875-8, page 621)
The ceremony is written so the lay person can be the presiding minister, for example:
“1662 A lay minister greets those present in the following words:
Praised by the God of grace, mercy, and peace. Blessed be God for ever.”
This varies from the instruction in the 1975 Roman Missal:
“The blessing and giving of ashes may be done outside Mass. In this case the entire liturgy of the word should be celebrated: entrance song, opening prayer, readings and chants, homily, blessing and giving of ashes, general intercessions.” (Roman Missal, Catholic Book Publishing Co., New York, 1985, page 78).
The 2002 Roman Missal seems to have the same thing. It could be argued that it overrules the Book of Blessings for the USA, since it is a more recent instruction. It has in Latin:
“Benedictio et impositio cinerum fieri potest etiam extra Missam. Quo in casu praemittitur Liturgia verbi, adhibendo cantum ad introitum, collectam, lectiones cum suis cantibus, ut in Missa. Sequitur deinde homilia et benedictio atque impositio cinerum. Ritus concluditur oratione universali, benedictione ac dimissione fidelium.”
(Missale Romanum, Libreria Editrice Vaticana, 2002, ISBN: 8820972719, page 200.)
The word “benedictio” means “blessing” – indicating that the ashes are to be blessed in the ceremony outside of Mass.