You must find it before going back inside the tower room, as you cannot head back outside after interacting with the radio.
Keep Flare Gun
If Emily keeps the gun for herself, she will have the option to either shoot or stow it. Shooting the flare gun will make it unusable in a later situation. Stowing it will be vital.
Later, while being chased in Chapter 8, if Emily dodged the upcoming Wendigo hands, went left while in the mines, threw off the stones, toppled both barrels, leaped off the conveyor belt or jumped over it, and has the flare gun, Emily will turn around and can shoot at the Wendigo approaching.
Successfully shooting it will allow her to escape without being bitten. Not being bitten gives Mike nothing to confront her about, which also spares her a potential death. Missing the shot will result in the Wendigo jumping on Emily and biting her. She will fall through the floor, and then run back to the lodge.
If Matt died at the cliff (making Emily alone in the fire tower) and Emily finds the flare gun, Emily will not have an option to give it to Matt because of his death, and rather the choice will be replaced from what would have happened if Emily kept the flare gun: shoot it or stow it.
Give Flare Gun To Matt
Giving the flare gun to Matt is an action shown in a Guidance totem. If Emily gives the flare gun to Matt, he will either shoot or stow it, depending on what he said about coming to the fire tower. Agreeing to come causes him to shoot it off immediately, which makes it unusable in a later situation, but disagreeing to come allows him to stow it away.
It can be used if Matt tries to save Emily both times (or just the second time) when the fire tower is falling. Attempting to save her puts him in a position where the Wendigo can find him and grab him. If he has the flare, he will remember it and aim it behind him. Successfully shooting the flare will cause the Wendigo to flee, and Matt will still be alive.
Not having the usable flare gun, or missing the shot, will result in Matt's death.
Not Having the Flare Gun
It is possible to not find the flare gun. The Butterfly Effect, Who Gets The Gun, will not appear. Neither characters will have a defense against their attacks, potentially opening up a greater chance of death for the two.
Weapons and Items