It may be a bug, Liquefaction. Many people have been having issues with the website since the summer sale began. If the button is the problem, my suggestions below may not be helpful, though they might be worth a try.
One option might be to use another browser and see if that makes a difference. I use Pale Moon most of the time, but it isn't working especially well with GOG's site right now, so I've been using Internet Explorer temporarily and haven't run into any problems.
Also, have you attempted to purchase your games through PayPal using your credit card? In many countries, you can do so without having or starting a PayPal account.
I believe you simply select the PayPal payment option when checking out on GOG and then make sure to click "Pay with Debit or Credit Card" at the bottom of the initial PayPal page. You then fill in your details and when PayPal is done processing your payment it links you back to GOG.
Be sure to uncheck the box on PayPal asking you if you want to create a new account (or save your personal info) at the bottom of the payment information page.
But, again, if it's the button on GOG's end that isn't working for you, that route likely isn't an option.
Hope you work it out (or it works itself out - or someone works it out). If you're still having trouble with the button over the next few hours, GOG support has always been very helpful and quick to respond in my experience.
Post edited June 11, 2017 by zakmckracken