As I understand the American usage, the abbeys of the American-Cassinese Congregation do not use the scapular. The abbeys of the Swiss-American Congregation do.
In the Benedictine tradition, there is no supernatural event associated with the scapular as the pious traditions present with the Carmelite, Dominican, and Norbertine scapulars among others. It is also not a prominent aspect of the monastic habit. For the Dominicans, for example, it is the scapular that is blessed when a Religious receives the habit. The scapular is the part of the habit deemed “quod essentium” (or the essential part) for them.
Relative to Benedictines, the scapular is a more practical and utilitarian part of the habit, serving the function of an apron to protect the tunic. That is not, however, to say that it does not retain its symbolic meaning of representing the “yoke of Christ” from the Gospel.
For an oblate (or a member of the English confraternity), a small scapular can be a miniature form of the habit that reminds and evokes the full monastic habit but in a manner that is suited to the life of the laity…there is no way to have a miniature version of the tunic or the cuculla, for example, that could be discreet and suitable. It is a sign of an affiliation to the Benedictine family through a specific house (or, for the confraternity, the English Congregation).
For those houses which retain the scapular, the investiture as an oblate novice is normally the moment for the clothing with the scapular.
Here is a newsletter from Saint Meinrad that features an investiture of an oblate novice: saintmeinrad.edu/media/228944/Oblate%20News%20Sp14.pdf
If you are interested in the confraternity, on the other hand, which is a different thing than being in oblate, you would need to contact one of the houses of the English Congregation for further information.
And, once again, an investiture in a scapular is to THAT scapular. Each scapular has a unique spirituality and expresses attachment to a unique family. The Carmelite scapular attaches you to Carmel and is an expression of Carmelite spirituality. Being invested with the Benedictine scapular has an entirely different purpose with an entirely different spirituality than the Carmelite scapular. A scapular is not an inter-changeable garment.
For the scapulars of the religious families (Carmelite, Dominican, Mercedarian, Norbertine, Servite, etc.), it is never a “stand alone” devotion like wearing a medal…it is indicative of a spirituality you are embracing and a relationship to the family of Religious, in some fashion, however attenuated. This is different from, for example, the scapular of Saint Joseph or the scapular of Saint Michael the Archangel.