Hello,
WoodMart theme has the option “Login to see add to cart and prices” in the Theme Settings > Shop. It means that the visitor should log-in or register.
You need to disable “Allow customers to place orders without an account” in Woocommerce > Settings > Account and Privacy. In addition, disable the automatic generation of the password and username.
You can configure the Registration form in this way: https://gyazo.com/e5c64556b28c782544e7d792487aa5b4
You can rewrite “Username” with Loco or search for a plugin that would add the fields you need.
Best Regards