You could create a form template – However, customizations are likely needed to validate and automatically add a user to a group.
You have Custom Wizard Plugin 🧙, but it’s unclear to me whether there is a validation system.
You can create a group with a custom template. However, that’s not super convenient to create a form within it. (That would be nice if you could use a Form Template here). It’s probably the closest to what you want.