When you first launch League of Legends, the first thing you want to do is go to your profile. From there you want to go to the champions tab. When you get to the champions tab, you want to select a random champion(it doesn't matter what champion you select). From there what you want to look for is whether or not you see "???" over the cost of the champion or the actual cost as in the number of IP and RP. If you see the cost of the champion you can go to your store, give it a second or two to load (it may take a bit) and then you should have full access to the shop without any Authentication errors or black screens. But if the same error pops up and says that "Authentication is Required" restart your league of legends client and try again. This may take a few times to work but it should show up eventually.
I don't know why this works but for whatever reason it works for me. Let me know how this goes for you and share this so that this error doesn’t bug anyone else.