Lock the App Drawer?

If you download a custom launcher you could either...

A. Remove the App Drawer entirely from that profile

OR

B. Hide certain apps so they don't show up even if the app drawer is pressed

Here is the link for the Launcher: Nova Launcher Prime

 

I've never tried it but you might look into Hide It Pro. It won't lock the app tray but you can hide apps.

Not sure if the stock launcher will let you do it, but with Apex (and probably other launchers) you could just drag the app tray shortcut to the trashcan so they can't use it. Or hide the dockbar completely and put shortcuts on the homescreen.

 

You can get the kid mode app, I use it and it's free