I don't think you can do this with the stock LG launcher, but you can install another launcher from the Google Play Store. I recommend Nova Launcer, it supports gestures. Another good alternative is Evie Launcher. Both of them support swiping up to open the app drawer.
That's exactly what I how I have my Nova set up. Swipe up for app launcher. Swipe down for notifications. Tap twice and swipe down for Waze. Two fingers swipe down to used to be Toggle Google Voice before it got updated.
I have Next 3D set up like that too. Subtle swipe up pulls up the app drawer. I also keep the dock hidden because I rarely need it, but a two finger swipe up and down will show and hide it when needed.