So this is the only way to delete your Gamestop account:
1 Call their customer service on 800-883-8895
that’s what they said in the very kind e-mail they’ve sent me.
Personally, I found it a bit weird that you need to use an offline device to close an online account, but on the other hand I understand that they do this for security reasons.
They might be asking you why you want to remove the account (e.g. because you’re simply not using it anymore) and run an identity check on you. After that, they will close the account for you.
Source (and thanks to): Customer service of Gamestop