The concept of a Domain is central to Windows, but not to Linux/UNIX. Emulating domain memberships on those operating systems is usually provided by add-on software. MIT Kerberos/OpenLDAP or Samba client is often used, as well as prepackaged offerings from VAS/QAS or Centrify.
Determining domain membership should be possible with any of these, but the methods would vary by product.
How do you determine their domain membership today (without BigFix)?