Method 1: Add the following function to your theme functions.php file:
If you don’t like editing your theme templates, or would like this functionality throughout all of your website themes, you should install the WP Email Login. That’s method 2 for log in in to WordPress using email address. I actually tested this plugin on a Multisite / BuddyPress installation yesterday and it works perfectly. You can download the WP Email Login plugin for free from the WordPress plugin repository.