Is it possible to add a user through the contact form in Shopify?
If you wanted to use the contact form as a registration form, one option would be to consider the opposite approach (using a registration form as a contact form).
Using an app like Customer Fields you can create custom registration forms and add them to whatever page you want. The form will add the user since it's a registration form, and you could add the normal contact fields to this form as well. It also has a feature that will send you alerts for any new submissions (like a contact form).