We have a number of different systems, that talk to each other. For every user, the user has a single username (email) and password. We can use the API to create a new customer and send them an activation email. This is not an issue.
How do we create a new customer that is marked as active, and set the password?
Checkout the documentation for the Customers API here. With regards to what you're asking, you can find how to do this under the Create a new customer with password and password_confirmation and skip sending the welcome email section.