I input my credit card details, but the green checkout button remains unselectable. I've tried Chrome, firefox and edge now, all with the same result.

Have you checked to make sure you've entered everything and that all the information is correct, including the name on the card?
If you're still having trouble, you might want to contact support.

Maybe post a screenshot of what the checkout screen looks like for you when this happens (omitting or redacting any sensitive info, of course). Chances are, it's something straightforward.

Can't check out? Welcome to the Hotel California. You can checkout any time you like, but you can never leave.

Next topic will be called "Can't post!"

Oh, wait, it won't.

(sorry, couldn't resist! Welcome to GOG forums!)

Welcome to the forums.

You should ask them nicely first before you do it.

Hehe, came for this; was not disappointed.

As for the OP probably contacting support is best.
Do you get any errors on the checkout page?
Have you got any browser add ons or extension that might interfere? (NoScript maybe?)

I thought this was a support forum, oops.

It is, we support each others flagging egos, but that doesn't help you.

It's not, but you're welcome to stay anyways. Come it, sit down, have a drink. Ignore the guy in the toga, he's been here for a while.

Bluto?

He'd fit in a toga?
Edit: Only now does it dawn on me that you aren't talking about the Bluto from Popeye.

Seeing that you are new you might make a mistake that has cought several newcomers here unaware: You can not buy DLCs to games that you do not have on GOG.

If for example you are trying to buy the DLC for The Witcher III without having the game on GOG that won't be possible without getting the game on your account first. If you bought the retail version of The Witcher III you should find a code somewhere in your box (usually on the back of the manual) which you can use to redeem the game to your account and only then can you continue to buy the DLCs.

If you aren't trying to buy a DLC simply disregard the above. But that is a possible reason why you are unable to buy something.

To be honest, I'd never even heard of it until I did a Google search to understand what reference you were making.