Objection 1. It would seem that a deacon, or one that is not a priest cannot grant an indulgence. Because remission of sins is an effect of the keys. Now none but a priest has the keys. Therefore a priest alone can grant indulgences.
Objection 2. Further, a fuller remission of punishment is granted by indulgences than by the tribunal of Penance. But a priest alone has power in the latter, and, therefore, he alone has power in the former.
On the contrary, The distribution of the Church's treasury is entrusted to the same person as the government of the Church. Now this is entrusted sometimes to one who is not a priest. Therefore he can grant indulgences, since they derive their efficacy from the distribution of the Church's treasury.
I answer that, The power of granting indulgences follows jurisdiction, as stated above (Question 25, Article 2). And since deacons and others, who are not priests, can have jurisdiction either delegated, as legates, or ordinary, as bishops-elect, it follows that even those who are not priests can grant indulgences, although they cannot absolve in the tribunal of Penance, since this follows the reception of orders.
This suffices for the Replies to the Objections, because the granting of indulgences belongs to the key of jurisdiction and not to the key of orders.
Source: Summa Theologica (New Advent)