You could build a signup just based on email addresses using the account recovery flow:
https://www.ory.sh/docs/kratos/manage-identities/account-recovery
So you create an account via the API with a random password, initiate the recovery flow for them with a custom email template
-> end-user gets a link via email which they can use to login and have to change their password.
(I think they would have to change the password upon first login, you could also make it optional to change later probably - but security-wise it would be best to change it directly IMO)