The member object doesn’t have user information directly; for that, you need to include the user (add a query param for include=user.null). In addition, you need to specify which fields to include (another query param fields[user]=first_name,last_name,... . See https://docs.patreon.com/#user-v2 for which fields can be requested on the user object and https://docs.patreon.com/#resources for an explanation of the JSON:API standard for requesting includes/fields.
The vanity field specifically is the user’s username
Hope this helps and let me know if you have further questions!