These are the specific terms with reference to religious priests:
Art. 2. In Masses celebrated without the people, each Catholic priest of the Latin rite, whether secular or regular, may use the Roman Missal published by Bl. Pope John XXIII in 1962, or the Roman Missal promulgated by Pope Paul VI in 1970, and may do so on any day with the exception of the Easter Triduum. For such celebrations, with either one Missal or the other, the priest has no need for permission from the Apostolic See or from his Ordinary.
Art. 3. Communities of Institutes of consecrated life and of Societies of apostolic life, of either pontifical or diocesan right, wishing to celebrate Mass in accordance with the edition of the Roman Missal promulgated in 1962, for conventual or “community” celebration in their oratories, may do so. If an individual community or an entire Institute or Society wishes to undertake such celebrations often, habitually or permanently, the decision must be taken by the Superiors Major, in accordance with the law and following their own specific decrees and statues.
ewtn.com/library/papaldoc/b16SummorumPontificum.htm
So religious priests have the same freedom as seculars with regard to private Mass, but there may be some restriction when it comes to regular public Masses
within the community. If done in a parish setting, it would seem there is once again no restriction. While many orders used to have their own rites and thus actually be restricted in certain ways from saying the Roman Rite, the past forty years have basically eliminated all of that distinctiveness so that it is easiest to think of religious priests as simply priests of the Roman Rite (and thus they can say both its forms).
