The short answer is yes. At least the Prior Provincial of the North American Carmelite Province of Elias says so.
There is a difference between wearing a blessed scapular and being enrolled in the scapular. Anyone may wear a blessed scapular if they wish. In fact, according to the Carmelite ritual book that provides the official enrollment, a person who “wears the scapular, but has no formal associate to the Order” still “shares in the graces traditionally associated with the Brown Scapular of Our Lady of Mount Carmel” (p. 3). Nevertheless, if one does wish to be formally enrolled in the Scapular of the Blessed Virgin Mary of Mount Carmel, it must be done in manner approved by the Carmelites themselves. (The Book of Blessings confirms this in paragraph #907.)
In the ritual book provided by the North American Provincials of the Carmelite Orders, however, there is an ambiguous line that seems to call into question whether any priest can enroll the faithful in the brown scapular. In the preliminaries, it says, “Priests and deacons have the faculties for blessing Scapulars; other authorized persons may enroll the faithful.”
Does this mean that, although priests and deacons may bless a scapular, they may not enroll the faithful unless explicitly authorized by the Carmelites? I asked this question to another priest. He reached out by email to a Carmelite Prior he knew and received this reply back:
And so, the Prior of the North American Carmelite Province of Elias (which is one of the five provinces that authorized this ritual book) affirms that any priest may enroll the faithful in the Carmelite brown scapular.
Here are a few photos of the ritual book referred to above:
