Yes they can, through several different methods.
You can enable the registration and login forms on the checkout page. Under Downloads > Settings > Payments > Checkout site owners have the option of including the login form, the registration form, or both on the checkout page.
When the above option is set to show both the login for and the registration form then checkout will look like the one below. Note the Login link at the top and the registration form at the bottom.
Anywhere On Your Site
If you’d like to allow your customers to log in at any time you can use the shortcode [edd_login] to render a login form. More documentation on [edd_login] here.
If you would like your customers to simply be able to create an account before purchasing or browsing the site you may use the shortcode [edd_register] and it will create a form for registration. See image below. More documentation on [edd_register] here.
Customer Account Page
For more information on creating a complete Customer Accout page, please read the Creating a Customer Account Page document.
Customer’s WordPress User Role
When a customer registers they are assigned to the WordPress default new user role. To verify or change go to the WordPress Settings > General and update “New User Default Role”.