If you want to close an Etsy account, you can easily do that in the Settings page of your account.
I’ll show you below with pictures how I managed to remove my profile (using a desktop computer):
1 Go to Etsy.com and sign in to your account
2 Select ‘You’ and then Account Settings in the dropdown menu
3 Scroll down to the bottom of the settings page
where you will see the button: ‘Close Account’
4 Enter your password and click on close account
Keep in mind it may take up to 30 minutes to delete your account.
You’ll be redirected to the homepage and in 30 minutes your account will be closed.
Source screenshots (and thanks to): etsy.com