It will mark it on your map even if you move the car.
EDIT 1: By put it in i don't mean just drop it into the car i mean put it in the trunk container.
EDIT 2: They can be bought at the barkeep and the scientist in the bunker in yantar.
https://www.reddit.com/r/stalker/comments/4aawm0/ogsepsa_put_a_gps_beacon_in_your_car/
The car is in the rookie village, south-east corner just next to the a house. Its a white car. Fuel can be found in the building near the grain mill (just before the bridge) and in the other structure you clear from bandits in one of the first missions. You can fuel the car by going to the back of the car with the fuel in your inventory(yes, the engine is in the back), pressing numlock 0 to open it up and numlock "." to fuel it. Get inside by pressing F near the door and press 0 to start the car once inside of it. Make sure to turn off the engine each time you get out (it doesn't always turn off when you get out).
https://www.reddit.com/r/stalker/comments/4d172r/cant_find_a_car_in_ogse/