You need the following -
1pr - Hands, Human
2pr - Gloves, Latex, Size Large
2ea - Tourniquet, Combat Application (or SOF-T, or anything that is PROVEN, RELIABLE, and which you can place ONE HANDED)
1ea - Combat Dressing, Izzy 6" with Slider (or an ace bandage and two rolls kerlix)
As a civilian patrol officer that is all you need. Ditch the hemostatic agents, if you simply must retain them to feel prepared, get one quickclot impregnated combat gauze roll and understand when and how to use it. If an officer has a minor penetrating injury to an extremity that you are fairly certain is not bleeding much ie. through and through gsw with no major vessel damage or minor stab wound with adipose tissue exposed bandage it with the izzy dressing. If the bleeding is any heavier, slap on the tourniquet and take him/her to the ER. If the officer has taken a hit in the thoracic area or abdominal area place a gloved hand over it and take them to the ER.
A tension pneumothorax is not going to develop before you can reach the hospital from blunt trauma after a vest strike. (If it does from a collision or something like that you probably wont be able to tell any way) If the pneumo is from an actual penetrating injury then you most likely will not need to needle decompress anyway as there is already a hole in the thoracic cavity to relieve pressure, the important thing is to prevent air from flowing into the pleural space thus the gloved hand over the wound.
mkmckinley - It is inappropriate for a civilian police officer with no medical training to insert a needle into another officers chest no matter how good their intentions, it makes no difference whether or not they will get in trouble for it.
PhilaPD - It is also very very ill advised for you to use hemostatic agents except in certain very specific instances. The most likely outcome of doing so is that you will significantly delay your buddies treatment at the hospital or significantly extend their surgical time prior to repairing the vasculature increasing their morbidity and mortality significantly. Dont do it.
Also thoracic injuries bleed but they typically dont bleed much if that makes sense, enough to be messy but not enough to matter physiologically unless something has been hit internally and there isnt anything you can do about that anyway.
Something else to consider is that you need to talk to your teammates and preplan. Not everyone wants to go by patrol car and if they want to wait for EMS you need to respect their wishes. Typically most of the thoracic injuries you will see are not a result of vest penetration but rather occur in the axillary region. These side penetration injuries have a huge risk of involving the spinal column made worse by the fact that most of these occur with a “bladed” stance rather than from a true 90 degree side angle and the projectile usually enters towards the front of the body and as it travels towards the center of the body it is angled rearward towards the spine. Many officers end up paralyzed as a result of this. The standard of care in the United States is to place a person with a likely spinal injury on a long backboard and do the best we can to immobilize that persons spinal column. If you drag that officer to the patrol car and pull him/her into the backseat with you, for better or worse their spinal column is not being protected. Most of us will agree that the damage is likely already done and no harm will come of this but there are officers who do not want to take that risk, you should talk about it and know ahead of time and respect their wishes.
Additionally you need to find a way to notify the hospital that you are coming in with a serious trauma patient so that they can activate the trauma team. Most hospitals will have people all over the place taking care of things and it take 3-5 minutes or longer to get everyone rounded up to adequately care for someone. Figure that part out before this happens, also if possible as part of your notification plan have the hospital staff meet you in the ambulance bay with a stretcher. Dragging a dying officer across the threshold of the ER trailing blood is not an effective way to calm the situation and ensure that officer is going to get effective initial care.
Place tourniquets, cover chest wounds, and drive fast. That is how you ensure your brothers and sisters have the best fighting chance possible to survive penetrating trauma in the line of duty and meet their responsibility to come home alive.