No. You do not require a Demat account to invest in mutual funds. Scripbox does not open one for you either. The record of mutual funds held by you is kept electronically in an account (Folio) maintained by the mutual fund company. You also receive regular statements from each of the mutual fund companies.
In addition, Scripbox maintains a record of your investments made through us and provides you a consolidated statement of all your holdings across all the mutual funds.