No, it is not necessary for a user who wishes to take part to have a Facebook account.
The administrator can decide whether or not users should be obliged to identify themselves via a Facebook account. In this case, at the moment of participation, the application will ask the user to begin a session in Facebook and accept the application permissions:
In the case of not requiring users to log in to Facebook in order to participate, users can complete the registration process without needing to be connected to Facebook.
Note: In this last case, we recommend that you always ask for the email in the registration form and that you enable the connection status bar.