Currently the website contact form can be modified using the widget form fields manager.
Standard form field submission rules apply to this facility. For more information read the custom form fields manual.
For the new field to appear in the contact form email template, please, edit the "admin_contact_form_submitted" template under "Administrator - Email - Email Templates" and add the field name pattern into the email text. For example, if you have added a new field and set its name to "companyType", add "{companyType}" pattern to the email template message body for the formatted field value to appear.