Is it possible to have a function run when a player respawns?

The CharacterAdded function. This triggers every time a Character is added. The event can be called from the Player model.

game.Players.LocalPlayer.CharacterAdded:Connect(function() -stuff- end)

or from a server script, as your localscript can reset too when the character respawns:

game.Players.PlayerAdded:Connect(function(p)
    p.CharacterAdded:Connect(function(c)
        --stuff
    end)
end)