You can add or remove emergency contacts from the Health app’s Medical ID on your iPhone with iOS 12, 13, and 14. Here’s how:
Open the Health app and tap the Medical ID tab.
Tap Edit, then scroll to Emergency Contacts.
Tap the + to add an emergency contact.
Tap a contact, then add their relationship.
Tap Done to save your changes.
Here’s how to remove emergency contacts:
Open the Health app and tap the Medical ID tab.
Tap Edit, then scroll to Emergency Contacts.
Tap the – (minus) next to a contact, then tap Delete.
Tap Done to save your changes.
(BTW, you can’t set emergency services as an emergency contact.)
(This how-to is based on my experiences and info on Apple’s support pages — where the images sometimes come from.)