Lets say I want to host a private instance that will not federate and I do not want to have a sign up button in the top right corner, only a login.

This private instance is a by invitation only, an exclusive club. Is there a way to create a form on another website or on the WWW subdomain of the domain for creating an account on the lemmy instance. This form will ask for a username and email for the lemmy account and before generating the account it can check to see if the name is available. And once the form is submitted, it will generate the account with a generated password and email the login details in an automated email to the user (username, password, URL to instance) and in the email inform the user to change their password when they login for good security measures?

Obviously the form will need to be programmed. Is there simple and secure API calls that one could use to do this and make a form with NodeJS or PHP to generate an account on lemmy?