There are 5 tools in The Last of Us. Finding all of them is necessary to upgrade your weapons. They are also a requirement for the "For emergencies only" gold trophy. Only finding the toolboxes won't earn you the trophy though. You still got to upgrade every weapon in the game (can be done at workbenches).
The tools have fixed locations and must be picked up in correct order during one playthrough. They do carry over to New Game+. Getting them via chapter select won't help with the upgrades or trophy!
#1 - Bill's Town (Graveyard) - 0:05
The first tool is in the very beginning of the Graveyard section, in the cellar where Bill shows you his selfmade Bomb.
#2 - Pittsburgh (Alone and Forsaken) - 0:30
The next toolbox is in the garage you enter in the beginning of the fifth chapter. It's right after you need to press Triangle to open the garage door.
#3 - The Suburbs (Sewers) - 0:52
This tool is after the area where you need to swim and get a wooden plank for Ellie. After activating the generator, Henry will help you out of the water. Now look in a side room on the left to find this toolbox on the ground.
#4 - The University (Science Building) - 1:25
This tool is located on the first floor of the science building. At the end of the corridor is a shiv door. You might need to craft a shiv first. Open the door and you will find the fourth tool.
#5 - Bus Depot (Highway Exit) - 1:55
The last of all tools is in a military tent in the outside area after watching the giraffes.