Hi I'm Brian and I'm here to help!
On a heat pump thermostat it is normal and proper for the outside unit (heat pump) to shut down when you physically switch your thermostat to emergency heat. That's what the emergency heat setting is for, it shuts of your outdoor unit and then your sole heating source becomes your backup heat (usually electric heat but sometimes gas or oil heat on a 'dual fuel' heat pump system.
If your backup heat is electric heating strips in your air handler (inside unit), then your electric heat strips should come on at that point. They should also come on automatically when you have your thermostat set in 'Heat' (not Emergency Heat) when the temperature in your home drops approximately 3 degrees below the heating setpoint on your thermostat.
With all that said, if your electric heat strips aren't coming on as stated above, you have a problem with them. Many times this is due to either blown fusible links in the heat strip or on occasion, and if you system is equipped with one, because there is an 'outdoor thermostat' that is locking out the heat strips when it is not cold enough to use them yet (usually set at about 35-40 degrees outdoor temperature). If your system is equipped with an outdoor thermostat and it has failed, it would keep your electric backup heat strips from running.
Diagnosing a problem with backup electric heat strips requires a volt/ohm meter and an amp-clamp ammeter. If you have those available and are comfortable using them on an energized circuit in your indoor unit then I can help you troubleshoot them further. I would need the make and model number of your indoor air handling unit to do so.
Let me know if you have further questions on this subject. Otherwise, please remember to rate before leaving (3 stars or higher please!). Bonuses are always appreciated! Thanks for allowing me to help and thanks for using Just Answer!
With all that said,