Personally, I think it’s a bit weird that you you need to use an offline device to close an online account, but it is what it is. This is the only way to delete your Walgreens account:
1 Call their customer service on 1-877-250-5823
and ask them to delete your online account. It’s toll-free and all you need to provide them is the email address that’s associated to the account.
They might be asking you why you want to remove the account (e.g. because you’re simply not using it anymore) and they’ll close the account immediately.
Source screenshots (and thanks to): http://www.walgreens.com/topic/accounthelp/account_help_main.jsp