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.

Alessandro Raffa

In some cases you want your wordpress users to login with their email address instead of the username. To do this, you can use a good plugin like ‘WP Email Login’ or you can activate a ‘Register Plus Redux’ plugin’s option, but when the users will ask for lost password, the old / original username will be emailed to them, and this can cause confusion. To avoid this situation, just install and activate Login Email Sync, and it will sync the username ( user_login database field ) with the email ( user_email database field ) every time a user updates her/his profile.
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Login Email Sync

In some cases you want your wordpress users to login with their email address instead of the username. To do this, you can use a good plugin like &#03 …

WooCommerce E-mail Login vs Login Email Sync