There are a couple of subtle nuances to this....
Registering a user is easy (two api calls, login and registeruser). However, that only adds the user and doesn't subscribe them to your group. They can't be subscribed (added to your group) until they confirm their email address. Thus, adding them as a user is somewhat useless.
There is a "directadd" api call that is great, but requires a paid premium subscription. I experimented with it yesterday and found it worked great. It adds the user to the group immediately (although they still have to create an account the first time they sign in)
In your case, what may make the most sense is to simply have them use an online form to sign up, from which you trigger a invitation to be sent (the invite endpoint). The invitation will give the user everything they need to create a password and join the group.