Enchiridion Indulgentiarum (sectio 11, page 58) offers a plenary indulgence for participating in Unity Week. You must attend aliquot functionibus (at least two public functions) and the concluding function; and, of course, fulfil the usual conditions.
A partial indulgence may be acquired by saying an approved prayer. The following* is on the back of a rather attractive prayer card issued in 1958 by the Anglo-Papalist Confraternity of Unity, on the occasion of the Golden Jubilee of the Chair of Unity Octave.
Antiphon That they all may be one, as Thou, Father, art in Me and I in Thee: that they also may be one in Us: that the world may believe that Thou hast sent Me.
V I say unto thee that thou art Peter.
R And upon this rock I will build My Church.
Let us pray.
O Lord Jesus Christ, Who saidst unto Thine Apostles: Peace I leave with you, My peace I give unto you: regard not our sins, but the faith of Thy Church, and vouchsafe to her that peace and unity which is according to Thy Will. Who livest and reignest God world without end. Amen*.
*Which includes the prayer accompanying the Pax in both the Ordinary and Extraordinary Forms, and so is certainly legitime adprobata. In the days of the Vetus Ordo, when this prayer was said silently by the Priest, it was not as commonly known among the laity as it is now.