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.
✅ Result
You’ll be redirected to the homepage and in 30 minutes your account will be closed.
Source screenshots (and thanks to): etsy.com