Do 13R or the 14 series get Combat Action Badges for actively engaging in the shooting down mortors or rockets?

13R don't shoot down mortars or rockets. 13R is a firefinder. They conduct counter battery missions. The use a radar to determine the point of origin of the attacking mortar, rocket, or artillery. Then they relay those coordinates to the 13J who relays the angle etc to the guns. The 13R never participate in any direct combat intentionally, and they don't fire anything.

I searched Rally Point and found a six year old answer from COL Jason Smallfield, PMP, CFM, CM which may provide some meaningful insight:
"According to ADRP 3-90 Offense and Defense (31 AUG 12), there are eight forms of contact including visual; direct; indirect; non-hostile; obstacles; aircraft; CBRN; and electronic warfare. According to AR 600-8-22 Military Awards (24 JUN 13), the Combat Action Badge may be awarded to any Soldier not eligible for the Combat Infantryman Badge (CIB) or Combat Medical Badge (CMB) after the date of September 18, 2001 performing duties in an area where hostile fire pay or imminent danger pay is authorized, who is personally present and actively engaging or being engaged by the enemy, and performing satisfactorily in accordance with the prescribed rules of engagement. Award of the CAB for indirect contact therefore falls within both doctrine and regulation. Commanders, however, should apply consistent criteria in awarding the CAB. Award of the CAB to a 2 mile long convoy engaged by direct fire only to the lead element, for example, would not be reasonable whereas an indirect attack where you were within the concussive wave and took cover would be reasonable in my opinion. Being in a barracks and not taking any action would not be appropriate. The keys are active engagement by the enemy and satisfactory performance IAW the regulation."

Very rarely we 14p will get a cab for shooting mortars down.