Why isn't my inventory system working?

You are checking if "a" is in the Player, while you have set a to clone the Flashlight. You should rather check if the Flashlight is in the player. I'm not sure if this will make a difference, but might as well try.

ALSO: You are only activating the function if the F key is pressed AND the Flashlight is in the character. If the Flashlight isn't there, the function isn't activating. This is what I suspect is the problem.

BP = script.Parent -- LocalScript is placed in the player's Backpack.
FL = BP.Flashlight
char = BP.Parent.Character -- Finds character.

Mouse = game.Players.LocalPlayer:GetMouse()

Mouse.KeyDown:connect(function(key)
    local a = FL:clone()
    if key == "f" then
    if char:FindFirstChild("Flashlight") then
        char:FindFirstChild("Flashlight"):Destroy() --remove is deprecated, use destroy.
    else
        a.Parent = char
    end
    end
end)

This should work, if not, let me know, and let me know if you have any errors in the output.