Can. 700 A decree of dismissal does not have effect unless it has been confirmed by the Holy See, to which the decree and all the acts must be transmitted; if it concerns an institute of diocesan right, confirmation belongs to the bishop of the diocese where the house to which the religious has been attached is situated. To be valid, however, the decree must indicate the right which the dismissed possesses to make recourse to the competent authority within ten days from receiving notification. The recourse has suspensive effect.
And then, according to the Commentary, the dicastery that deals with it at the Vatican is the Congregation for Institutes of Consecrated Life and Societies of Apostolic Life, with the appeal going to the Apostolic Signatura.
Thank you, markomalley! Canon Law marches on.