I am trying to set up contact forms for individual members of staff but in the notification settings I only have the option of sending an email to myself, the site administrator.  How do I make it possible for other users to receive email notification upon form submission?

You can only setup notifications for those that have permission to carry out the task e.g.you'd need the "documents & media" permission on the group where the article is contained as that permission is required to view the results that are posted to an article form. If you don't see anyone other than yourself listed then it's because you are the only one that has permission to access that article in the Web Office.

When it comes to article forms it has been pointed out that it would be useful to add other users to receive the email notification, even if they don't have the permission. We haven't added this yet but feel free the add a feature request to raise it up the priority list.

In the meantime, you can workaround the issue by creating a new user for each person you wish to add to the email list. These users could be created in a 'FORM NOTIFICATION USERS' mailing list group. Don't give the user login access but do add their email address. You can then assign the user the 'documents and media' permission in the relevant group where the form is located and then they'll be available for selection for form notifications.