Yes, it can be done, and often was before GPS.
Years ago, I was an Army helo pilot. Did my IFR ride in the helicopter with a single Nav receiver and NDB, no DME or GPS. Holding over an airway intersection with a single VOR, Localizer approach with step downs, NDB approach, 0/0 instrument take off, the works and all without an autopilot. The ILS was the easy part. Now THAT was a busy check ride! I was much younger then, wouldn’t want to do it today.
As others have said, the trick is finding the right instructor and the right DPE. I think one would have to find “old school” types to get it done. I would remove the portable mounted in the panel so as to force you to learn the “old school” way AND to prevent you from using an unapproved device for navigation on the checkride.