By default, you and your customers will have to login using your username. This is not practicall since WooCommerce will auto-generate usernames for your customers and they will often forget what their username is. This plugin will resolve that by enabling your users to login using their e-mail address.
<ul>
<li>Works with WooCommerce</li>
<li>Perfect with Klarna Checkout</li>
</ul>

WooCommerce E-mail Login

Enables you and your customers to login with your e-mail address as well as your username.

Brian Henry

A new “Email Magic Link” button is added to the standard WordPress and WooCommerce login screens. If there is a user 
account for the username filled out, they will receive an email with a link to log them in without a password.
All emails sent from WordPress will contain login codes in links pointing back to the website:
<ul>
<li>Comment reply emails</li>
<li>Abandoned cart emails</li>
<li>Membership reminder emails</li>
<li>etc.</li>
</ul>
No configuration is required, by default:
<ul>
<li>Autologin URLs work for one week</li>
<li>Emails to administrators are excluded</li>
<li>Emails on exclusion shortlist are not modified</li>
</ul>
An API is available for developers to use autologin codes elsewhere in WordPress, e.g. push notifications, and to conditionally disable the plugin’s use. Code is published on GitHub, uses WordPress Plugin Boilerplate, conforms (mostly) to WordPress Coding Standards, and is unit & integration tested.

Magic Emails & Autologin URLs

Adds magic email link to login screen. Adds single-use passwords to WordPress emails' URLs for frictionless login.

WooCommerce E-mail Login vs Magic Emails & Autologin URLs