Using the latest 1.6 patch (with all 3 DLCs):

Seems to be a bug for the GOG version, clicking the white gem opens up that small window where you can click 'purchaseables' or 'use', but when you do it just closes the window like nothing happened.

Is there really no way for GOG customers to buy the cosmetics? You can't do it on the main website, either...

I'm in online mode, my account is connected, everything is running fine. It's just the cosmetic shop won't work. :(

I don't know if this is related, but I kept getting Lua Errors while trying to start up an old toon.
The error included something about cosmetics in the line as well.

Just adding info.

This came up in game chat today, from what I recall of it darkgod said that GOG wouldn't allow the shop in the GOG release. You can download the client from the official site and log in with the same account for the purchases, apparently.

A recommendation was made to add it to the official site, and IIRC darkgod said he'll look into it.