General assumptions: Stealing Silverballers is a better idea if you are not good at the shooting range, especially that retrieving Hitman's pistols is possible to achieve without attacking local people.
In order to steal Silverballers you have to obtain a key to the cabinet. It is stored in the gun shop owner's safe. The only way to open the safe is obtaining a notebook with right safe combination and you can do it in two ways. The first method is finding a notebook in the survival bunker. Start with going to the right. Look around in the main gun shop room, finding a fuse box shown on the above screen. Sabotage the box when no one looks at you.
Return now to the starting place, but do not run not to attract unnecessary attention. Wait until the guard standing next to the counter will go to check the fuse box.
When playing on lower difficulty levels you can go behind the counter right away and take a keycard. On higher difficulty levels you may encounter a second guard there, so it is better to wait until he leaves this place.
The next step is using the right door leading to the back of the gun store. If you play on higher difficulty levels you have to watch out for another guard and in this case you should wait until he goes upstairs. When you get to the back of the store, select the right corridor and locate the door shown on the above screen.
Crouch and turn left, getting into trenches. Follow the linear path here, stopping only after reaching a bunker shown on the above screen. Use the recently acquired keycard to unlock the door.
Inside the bunker you'll find first of all a notebook with the safe combination. Look around to find also evidence. After picking up both these items return to the door leading to the back of the gun shop, but do not use it yet.
Go to the left, sneaking near sandbags, at which persons on shooting range stand. Stop at the place shown on the above screen. On lower difficulty levels you can just jump between covers to reach the left building. On higher difficulty levels you can be exposed in this way so it is better to wait until a guard visible in a distance leaves his stand (he will try to reprimand Lynch).
Regardless of how you get close to gun shop owner's office, use the shutter shown on the above screen, but first make sure that the room is empty. The only thing left now is walking to the safe and opening it.
The second option is to take away the notebook from the gun shop owner. I strongly advise not to attacking this person in public place, because you would alert all local characters. Instead of that, go to the right. Use the side passage leading to the shooting range (screen above).
Stop at the place, where few policemen shoot ducks and examine the right wall in order to find an entrance to the ventilation shaft (screen above).
Go through the tunnel to find yourself in the office. Hide in the closet and wait until the gun shop owner comes in. Do not leave the closet until he stops at the desk, turning back to you.
Only now exit the closet. Carefully approach the gun shop owner and depending on your preferences stun or kill him. Hide the body and look around for the notebook lying on the floor. Now go to the nearby safe and open it.
Regardless of your behavior, you have to open the safe and take a keycard to the cabinet out of it. It is the safest to leave the office by the ventilation shaft you used to get here.