Launchpad keeps rearranging icons
So I just tested this out on my computer by moving around some icons. Looks like the file that stores the information is here:

~/Library/Preferences/com.apple.LaunchServices.plist and if you have it on your system ~/Library/Preferences/com.apple.LaunchServices.QuarantineEventsV2

So to test you can grab the two files and drag them over to the desktop. I would suggest logging out and back in of your account and then going back into Launch pad to test if issue is resolved.