When I am playing games on my game controller and I push L3 it brings up the virtual keyboard. It is very annoying to be in a game and that pop up. I can not figure out how to change the keybinds for this. Can someone help?

GOG Galaxy doesn't have anything to do with your controller bindings. These are set in the game itself, controller driver software, or other launchers like Steam. The latter running in the background should be the first thing to check.

A cursory search for L3 controller virtual keyboard suggests it is typically the water vapour store client (i.e. Steam) that causes the problem.

Do you have that client running? If so, do the following on the water vapour client and see if it helps. Searching the water vapour client forums can provide more details, but I'm forbidden from linking it here.